452 results
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2. On D.Y. Gao and X. Lu paper 'On the extrema of a nonconvex functional with double-well potential in 1D'
- Author
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Constantin Zălinescu
- Subjects
021103 operations research ,Applied Mathematics ,General Mathematics ,0211 other engineering and technologies ,General Physics and Astronomy ,Double-well potential ,02 engineering and technology ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Maxima and minima ,35J20, 35J60, 74G65, 74S30 ,Optimization and Control (math.OC) ,FOS: Mathematics ,Preprint ,0101 mathematics ,Constant (mathematics) ,Mathematics - Optimization and Control ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3
- Published
- 2017
3. Some comments on the paper: Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959
- Author
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Michelle Pierri and Donal O'Regan
- Subjects
Discrete mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Differential systems ,01 natural sciences ,010101 applied mathematics ,Controllability ,Algebra ,0101 mathematics ,Differential (mathematics) ,Mathematics - Abstract
The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.
- Published
- 2016
4. Remark on a paper of R. Usha and K. Prema concerning a linear parabolic system
- Author
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René P. Sperb
- Subjects
Parabolic system ,Pure mathematics ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Mathematics - Published
- 1999
5. Minimization arguments in analysis of variational-hemivariational inequalities
- Author
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Weimin Han and Mircea Sofonea
- Subjects
Applied Mathematics ,General Mathematics ,Hilbert space ,Structure (category theory) ,General Physics and Astronomy ,Contrast (statistics) ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Contact mechanics ,Compact space ,symbols ,Applied mathematics ,Minification ,0101 mathematics ,Mathematics - Abstract
In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.
- Published
- 2022
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6. Stability and instability results for Cauchy laminated Timoshenko-type systems with interfacial slip and a heat conduction of Gurtin–Pipkin’s law
- Author
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Aissa Guesmia
- Subjects
Polynomial ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Cauchy distribution ,Dissipation ,Type (model theory) ,Thermal conduction ,symbols.namesake ,Thermoelastic damping ,Fourier analysis ,Law ,symbols ,Variable (mathematics) ,Mathematics - Abstract
The subject of the present paper is to study the stability of a class of laminated Timoshenko-type systems in the whole line $$\mathbb {R}$$ combined with a heat conduction given by Gurtin–Pipkin’s law and acting only on one equation of the laminated Timoshenko-type system. The main result of this paper shows that the thermoelastic dissipation generated by Gurtin–Pipkin’s law is strong enough to stabilize the system at least polynomially, even if only the second or the third equation of the laminated Timoshenko-type system is controlled and the two other ones are free. When only the first equation of the laminated Timoshenko-type system is controlled, we give a necessary and sufficient condition for the polynomial stability. The polynomial decays in the $$L^2$$ -norm of the solution, and its higher-order derivatives with respect to the space variable are specified in terms of the regularity of the initial data and some connections between the coefficients. An application to the particular case of Timoshenko-type systems is also given. The proofs are based on the energy method and Fourier analysis combined with some well-chosen weight functions.
- Published
- 2021
7. Local and global existence of mild solutions of time-fractional Navier–Stokes system posed on the Heisenberg group
- Author
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Djihad Aimene, Djamila Seba, and Mokhtar Kirane
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Time derivative ,Heisenberg group ,General Physics and Astronomy ,Order (ring theory) ,Fixed-point theorem ,Development (differential geometry) ,Navier stokes ,Uniqueness ,Navier–Stokes equations ,Mathematics - Abstract
This paper is a development of the results and techniques of the two papers (Carvalho-Neto and Planas in J Differ Equ 259:2948–2980, 2015; Oka in J Math Anal Appl 473:382–407, 2019) for the aim of addressing the existence and uniqueness of local and global mild solutions, on the Heisenberg group $$\mathbb {H}^d$$ , of the time-fractional Navier–Stokes system with time derivative of order $$\alpha \in (0, 1)$$ . The proof relies on Schaefer’s fixed point theorem.
- Published
- 2021
8. Solution for nonvariational quasilinear elliptic systems via sub-supersolution technique and Galerkin method
- Author
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Leandro S. Tavares, Francisco Julio S. A. Corrêa, and Gelson C. G. dos Santos
- Subjects
Change of variables ,Elliptic systems ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Monotonic function ,Type (model theory) ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,symbols.namesake ,symbols ,Dissipative system ,Applied mathematics ,0101 mathematics ,Galerkin method ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we obtain the existence of positive solution for a system of quasilinear Schrodinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrodinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrodinger equations.
- Published
- 2021
9. Well-posedness and asymptotic behaviour of a wave equation with non-monotone memory kernel
- Author
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Genqi Xu and Rongsheng Mu
- Subjects
Lyapunov function ,Semigroup ,Function space ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Monotonic function ,Function (mathematics) ,symbols.namesake ,Monotone polygon ,Exponential stability ,Kernel (statistics) ,symbols ,Applied mathematics ,Mathematics - Abstract
In this paper, we study the well-posedness and stability of a wave equation with infinitely structural memory, herein the memory kernel function does not satisfy the monotonicity. For the model, the history function space setting is a main difficulty because the usual space setting will lead the shift semigroup to be a unbounded semigroup. In the present paper, we modify the history function space setting and prove the well-posedness of the system. Further we study the stability of the system via Lyapunov function method. By constructing appropriate Lyapunov function, we show that the energy function of the system decays exponentially if the memory kernel function satisfies some conditions. Finally, we give an example of the memory kernel function that is not monotone but satisfies all conditions proposed in the present paper.
- Published
- 2021
10. Double phase problems with variable growth and convection for the Baouendi–Grushin operator
- Author
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Vicenţiu D. Rădulescu, Anouar Bahrouni, and Patrick Winkert
- Subjects
Convection ,Partial differential equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Differential operator ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Nonlinear system ,Operator (computer programming) ,0101 mathematics ,Transonic ,Variable (mathematics) ,Mathematics - Abstract
In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.
- Published
- 2020
11. Periodic traveling wavefronts of a multi-type SIS epidemic model with seasonality
- Author
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Haiqin Zhao and Yumeng Gu
- Subjects
Wavefront ,education.field_of_study ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Population ,General Physics and Astronomy ,Type (model theory) ,Stability (probability) ,Coincidence ,Distribution (mathematics) ,Uniqueness ,Epidemic model ,education ,Mathematics - Abstract
This paper is concerned with a time-periodic and nonlocal system arising from the spread of a deterministic epidemic in multi-types of population by incorporating a seasonal variation. The existence of the critical wave speed of the periodic traveling wavefronts and its coincidence with the spreading speed were proved in Wu et al. (J Math Anal Appl 463:111–133, 2018). In this paper, we prove the uniqueness and stability of all non-critical periodic wavefronts. Of particular interest is the influences of time-periodicity on the spreading speed in one-dimensional case. It turns out that, in comparison with the autonomous case, the periodicity of the infection rate increases the spreading speed, while the periodicity of the combined death/emigration/recovery rate for infectious individuals decreases the spreading speed. We also find that the contact distribution increases the spreading speed.
- Published
- 2020
12. On the global regularity for a 3D Boussinesq model without thermal diffusion
- Author
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Weinan Wang
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Uniqueness ,Persistence (discontinuity) ,Thermal diffusivity ,Mathematics - Abstract
In a recent paper (Ye in Z Angew Math Phys 68:83, 2017), Ye proved the global persistence of regularity for a 3D Boussinesq model in $$H^{s}({\mathbb {R}}^3) \times H^{s}({\mathbb {R}}^3)$$ with $$s>5/2$$. In this paper, we show that the global persistence and uniqueness still hold when $$s>3/2$$.
- Published
- 2019
13. Viscoelastic versus frictional dissipation in a variable coefficients plate system with time-varying delay
- Author
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Jianghao Hao and Peipei Wang
- Subjects
0209 industrial biotechnology ,Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Ode ,General Physics and Astronomy ,02 engineering and technology ,Function (mathematics) ,Dissipation ,01 natural sciences ,Viscoelasticity ,Exponential function ,020901 industrial engineering & automation ,Relaxation (physics) ,0101 mathematics ,Variable (mathematics) ,Mathematics - Abstract
In this paper, we are concerned with variable coefficients plate system subjected to three partially distributed feedbacks: time-varying delay, frictional and viscoelastic dissipations. This work is devoted to, without any prior quantification of both decay rate of relaxation function and growth rate of frictional dissipation near the origin, establish a general decay result which corresponds to a certainly stable ODE. Our result extends the decay result obtained for some kind of problems with finite history to problem with infinite history. Moreover, this paper allows a wider class of kernels of infinite history, and the usual exponential and polynomial decay rates are only special cases. The proof is based on the multiplier method and some techniques about convex functionals.
- Published
- 2019
14. Optimal control for a class of mixed variational problems
- Author
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Mircea Sofonea, Yi-bin Xiao, and Andaluzia Matei
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Weak formulation ,Lipschitz continuity ,Optimal control ,Strongly monotone ,01 natural sciences ,010101 applied mathematics ,Mosco convergence ,Compact space ,Applied mathematics ,Uniqueness ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The present paper concerns a class of abstract mixed variational problems governed by a strongly monotone Lipschitz continuous operator. With the existence and uniqueness results in the literature for the problem under consideration, we prove a general convergence result, which shows the continuous dependence of the solution with respect to the data by using arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Then we consider an associated optimal control problem for which we prove the existence of optimal pairs. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to mixed variational problems. To provide an example, we illustrate our results in the study of a mathematical model which describes the equilibrium of an elastic body in frictional contact with a foundation.
- Published
- 2019
15. Two-dimensional strain gradient damage modeling: a variational approach
- Author
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Emilio Barchiesi, Anil Misra, and Luca Placidi
- Subjects
Karush–Kuhn–Tucker conditions ,Deformation (mechanics) ,Applied Mathematics ,General Mathematics ,Linear elasticity ,Mathematical analysis ,Isotropy ,General Physics and Astronomy ,02 engineering and technology ,Dissipation ,01 natural sciences ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Damage mechanics ,0101 mathematics ,Galerkin method ,Energy functional ,Mathematics - Abstract
In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed.
- Published
- 2018
16. Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger–Moser type
- Author
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João Marcos do Ó and Jianjun Zhang
- Subjects
Standing wave ,symbols.namesake ,Nonlinear system ,Applied Mathematics ,General Mathematics ,symbols ,General Physics and Astronomy ,Geometry ,Type (model theory) ,Schrödinger equation ,Mathematical physics ,Mathematics - Abstract
In this paper, we deal with the following singularly perturbed elliptic problem $$\begin{array}{ll}-\varepsilon^2\Delta u+V(x)u=f(u),\quad u \in H^1(\mathbb{R}^2),\end{array}$$ where f(s) has critical growth of Trudinger–Moser type. In this paper, we construct a localized bound-state solution concentrating at an isolated component of the positive local minimum points of V as $${\varepsilon \rightarrow 0}$$ under certain conditions on f(s). Our results complete the analysis made in Byeon et al. (Commun Partial Differ Equ 33: 1113–1136, 2008) for the two-dimensional case, in the sense that, in that paper only the subcritical growth was considered.
- Published
- 2015
17. Cauchy problems of pseudo-parabolic equations with inhomogeneous terms
- Author
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Zhongping Li and Wanjuan Du
- Subjects
Life span ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Fujita exponent ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Cauchy distribution ,Infinity ,Parabolic partial differential equation ,Term (time) ,Initial value problem ,Critical exponent ,media_common ,Mathematics - Abstract
This paper deals with Cauchy problems of pseudo-parabolic equations with inhomogeneous terms. The aim of the paper is to study the influence of the inhomogeneous term on the asymptotic behavior of solutions. We at first determine the critical Fujita exponent and then give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity. Furthermore, the precise estimate of life span for the blow-up solution is obtained. Our results show that the asymptotic behavior of solutions is seriously affected by the inhomogeneous term.
- Published
- 2015
18. Spectrum and stability analysis for a transmission problem in thermoelasticity with a concentrated mass
- Author
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Genqi Xu and Zhong-Jie Han
- Subjects
Multiplier (Fourier analysis) ,Thermoelastic damping ,Transmission (telecommunications) ,Applied Mathematics ,General Mathematics ,Frequency domain ,Mathematical analysis ,Spectrum (functional analysis) ,General Physics and Astronomy ,Exponential decay ,Stability (probability) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, a transmission problem between elastic and thermoelastic material is considered. Assume that these two materials are connected by a vibrating concentrated mass. By a detailed spectral analysis, the asymptotic expressions of the eigenvalues of the system are obtained, and based on which, the Riesz basis property of the eigenvectors is deduced. It is proved that the total energy of this system cannot achieve exponential decay. However, by the frequency domain method together with some multiplier techniques, the polynomial decay of the system is showed and the optimal decay rate is estimated. Finally, some numerical simulations are given to support the results obtained in this paper.
- Published
- 2015
19. Periodic solutions to nonlinear wave equations with spatially dependent coefficients
- Author
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Jinhai Chen
- Subjects
Inverse function theorem ,Nonlinear system ,Nonlinear wave equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Spectrum (functional analysis) ,General Physics and Astronomy ,Boundary value problem ,Uniqueness ,D'Alembert operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper investigates the existence and uniqueness of weak solutions to a periodic boundary value problem for a system of nonlinear wave equations with spatially dependent coefficients. Priori estimates of weak solutions are also established for the periodic boundary value problem. The arguments rely on spectral properties of the corresponding wave operator and a global inverse function theorem. The results presented in this paper extend the ones known in the literature in that eigenvalues of nonlinear perturbing terms appeared in the system of nonlinear wave equations can be chosen from the spectrum of the underlying wave operator.
- Published
- 2015
20. Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant
- Author
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Chunlai Mu, Ke Lin, Liangchen Wang, and Jie Zhao
- Subjects
Combinatorics ,Homogeneous ,Applied Mathematics ,General Mathematics ,Domain (ring theory) ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Neumann boundary condition ,General Physics and Astronomy ,Nabla symbol ,Diffusion function ,Omega ,Mathematics - Abstract
This paper deals with an initial-boundary value problem for the chemotaxis system $$\left\{\begin{array}{ll} u_t = \nabla \cdot (D (u) \nabla u)- \nabla \cdot (u \nabla v), \quad & x\in \Omega, \quad t > 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t > 0, \end{array}\right.$$ under homogeneous Neumann boundary conditions in a convex smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) with \({n\geq3}\), where the diffusion function D(u) satisfying $$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u > 0 \end{array}$$ with some cD > 0 and m > 1. The main goal of this paper was to extend a previous result on global existence of solutions by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014) under the condition that \({m > 2-\frac{2}{n}}\) can be relaxed to \({m > 2-\frac{6}{n+4}}\).
- Published
- 2015
21. Higher-order asymptotic result for the wrinkling of an everted Varga spherical shell
- Author
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N. Abdolalian and M. Sanjaranipour
- Subjects
Current (mathematics) ,Buckling ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Ode ,General Physics and Astronomy ,Critical radius ,Eigenfunction ,Asymptotic expansion ,Spherical shell ,WKB approximation ,Mathematics - Abstract
This paper applies the WKB method to the buckling analysis of an everted spherical shell composed of Varga material. The same problem has been studied by Haughton and Chen in 2003, but they only obtained the leading-order value for the critical radius ratio B/A because they did not seem to realize that the second-order ODE with variable coefficients satisfied by the eigenfunctions could be solved explicitly, where A and B are the inner and outer radii of the undeformed sphere. In the current paper, we managed to find the leading-order and next-order eigenfunctions explicitly in the WKB expansion, and hence obtained two more terms in the asymptotic expansion of B/A. We believe that the same idea can be employed to find the eigenfunctions at higher orders.
- Published
- 2014
22. Global existence and finite time blow-up for a class of thin-film equation
- Author
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Zhihua Dong and Jun Zhou
- Subjects
Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,01 natural sciences ,Upper and lower bounds ,010101 applied mathematics ,Nonlinear system ,Thin-film equation ,0101 mathematics ,Finite time ,Constant (mathematics) ,Mathematics - Abstract
This paper deals with a class of thin-film equation, which was considered in Li et al. (Nonlinear Anal Theory Methods Appl 147:96–109, 2016), where the case of lower initial energy ( $$J(u_0)\le d$$ and d is a positive constant) was discussed, and the conditions on global existence or blow-up are given. We extend the results of this paper on two aspects: Firstly, we consider the upper and lower bounds of blow-up time and asymptotic behavior when $$J(u_0)d$$ .
- Published
- 2017
23. Global regularity for a 3D Boussinesq model without thermal diffusion
- Author
-
Zhuan Ye
- Subjects
010101 applied mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Compressibility ,General Physics and Astronomy ,0101 mathematics ,Boussinesq approximation (water waves) ,Thermal diffusivity ,01 natural sciences ,Heat kernel ,Mathematics - Abstract
In this paper, we consider a modified three-dimensional incompressible Boussinesq model. The model considered in this paper has viscosity in the velocity equations, but no diffusivity in the temperature equation. To bypass the difficulty caused by the absence of thermal diffusion, we make use of the maximal $$L_t^{p}L_x^{q}$$ regularity for the heat kernel to establish the global regularity result.
- Published
- 2017
24. Existence of nontrivial solution for Schrödinger–Poisson systems with indefinite steep potential well
- Author
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Juntao Sun, Yuanze Wu, and Tsung-fang Wu
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Lambda ,Poisson distribution ,01 natural sciences ,Omega ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$30$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .
- Published
- 2017
25. Optimal control of rigidity parameters of thin inclusions in composite materials
- Author
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Luisa Faella, Alexander Khludnev, and Carmen Perugia
- Subjects
genetic structures ,General Mathematics ,Rigid inclusion ,General Physics and Astronomy ,Geometry ,02 engineering and technology ,01 natural sciences ,Thin inclusion ,Control function ,Thin inclusion, Rigid inclusion, Optimal control, Elastic body, Crack, Nonpenetration condition ,0203 mechanical engineering ,Equilibrium problem ,Boundary value problem ,0101 mathematics ,Mathematics ,Crack ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Existence theorem ,Optimal control ,Nonpenetration condition ,Elastic body ,020303 mechanical engineering & transports ,Displacement field - Abstract
In the paper, an equilibrium problem for an elastic body with a thin elastic and a volume rigid inclusion is analyzed. It is assumed that the thin inclusion conjugates with the rigid inclusion at a given point. Moreover, a delamination of the thin inclusion is assumed. Inequality type boundary conditions are considered at the crack faces to prevent a mutual penetration between the faces. A passage to the limit is justified as the rigidity parameter of the thin inclusion goes to infinity. The main goal of the paper is to analyze an optimal control problem with a cost functional characterizing a deviation of the displacement field from a given function. A rigidity parameter of the thin inclusion serves as a control function. An existence theorem to this problem is proved.
- Published
- 2017
26. Equivalence between observability at the boundary and stabilization for transmission problem of the wave equation
- Author
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A. J. A. Ramos and M. W. P. Souza
- Subjects
0209 industrial biotechnology ,Wave propagation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Boundary (topology) ,02 engineering and technology ,Wave equation ,01 natural sciences ,Domain (mathematical analysis) ,020901 industrial engineering & automation ,Exponential stability ,Observability ,0101 mathematics ,Hyperbolic partial differential equation ,Equivalence (measure theory) ,Mathematics - Abstract
In this article, we have studied the transmission problem of a system of hyperbolic equations consisting of a free wave equation and a wave equation with dissipation on the boundary, each one acting on a part of its one-dimensional domain. This paper proves the equivalence between the exponential stability previously proven by Liu and Williams (Bull Aust Math Soc 97:305–327, 1998) and the inequality observability on the boundary as a result of this paper. First of all, we have built an auxiliary problem on where we extracted some slogans to be used later. Then we have introduced a number $${\mathcal {X}}>0$$ representing the difference between the speed of wave propagation in each part of the domain, and we proved one observability inequality on the boundary. Finally, we proved the equivalence between the two properties.
- Published
- 2017
27. L p -convergence rates to nonlinear diffusion waves for quasilinear equations with nonlinear damping
- Author
-
Lina Zhang and Shifeng Geng
- Subjects
Nonlinear system ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Convergence (routing) ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Nonlinear diffusion ,Hyperbolic partial differential equation ,Mathematics - Abstract
This paper is concerned with the asymptotic behavior of the solution for quasilinear hyperbolic equations with nonlinear damping. The main novelty in this paper is that we obtain the Lp(2 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations, and we need none of the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton (Q Appl Math 61:295–313, 2003).
- Published
- 2013
28. Thin-film superconducting rings in the critical state: the mixed boundary value approach
- Author
-
Francesco Grilli and Roberto Brambilla
- Subjects
Superconductivity ,Washer ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Geometry ,STRIPS ,Boundary values ,law.invention ,Magnetic field ,law ,Boundary value problem ,Thin film ,Complex plane ,Mathematics - Abstract
In this paper, we describe the critical state of a thin superconducting ring (and of a perfectly conducting ring as a limiting case) as a mixed boundary value problem. The disc is characterized by a three-part boundary division of the positive real axis, so this work is an extension of the procedure used in a previous work of ours for describing superconducting discs and strips, which are characterized by a two-part boundary division of the real axis. Here, we present the mathematical tools to solve this kind of problems—the Erdelyi–Kober operators—in a frame that can be immediately used. Contrary to the two-part problems considered in our previous work, three-part problems do not generally have analytical solutions and the numerical work takes on a significant heaviness. Nevertheless, this work is remunerated by three clear advantages: firstly, all the cases are afforded in the same way, without the necessity of any brilliant invention or ability; secondly, in the case of superconducting rings, the penetration of the magnetic field in the internal/external rims is a result of the method itself and does not have to be imposed, as it is commonly done with other methods presented in the literature; thirdly, the method can be extended to investigate even more complex cases (four-part problems). In this paper, we consider the cases of rings in uniform field and with transport current, with or without flux trapping in the hole and the case without net current, corresponding to a cut ring (washer), as used in some SQUID applications.
- Published
- 2013
29. Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations
- Author
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Giuseppe Saccomandi and Kumbakonam R. Rajagopal
- Subjects
Class (set theory) ,Wave propagation ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Context (language use) ,Function (mathematics) ,Viscoelasticity ,Physics::Fluid Dynamics ,Stress (mechanics) ,Classical mechanics ,Shear stress ,Tensor ,Mathematics - Abstract
In this paper, we show that circularly polarized transverse stress waves, standing shear stress waves, and oscillatory shear stress waves can propagate in a new class of viscoelastic solid bodies which are a subclass of bodies described by implicit constitutive theories. The class of models that is being considered includes as sub-classes, the classical Kelvin–Voigt model, the new models introduced by Rajagopal wherein the Cauchy–Green tensor is a non-linear function of the stress, and the Navier–Stokes fluid model. The solutions established in this paper are generalizations of solutions that have been established within the context of nonlinear elasticity by Carroll, and Destrade and Saccomandi, to the new class of elastic and viscoelastic bodies that are being considered.
- Published
- 2013
30. Quasilinear parabolic variational inequalities with multi-valued lower-order terms
- Author
-
Vy Khoi Le and Siegfried Carl
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Regular polygon ,Solution set ,General Physics and Astronomy ,Function (mathematics) ,Domain (mathematical analysis) ,Combinatorics ,Elliptic operator ,Directed set ,Obstacle problem ,Variational inequality ,Mathematics - Abstract
In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.
- Published
- 2013
31. Finite time blow-up for a reaction-diffusion system in bounded domain
- Author
-
Xueli Bai
- Subjects
Applied Mathematics ,General Mathematics ,Open problem ,Mathematical analysis ,Null (mathematics) ,General Physics and Astronomy ,Domain (mathematical analysis) ,Dirichlet distribution ,Parabolic system ,symbols.namesake ,Bounded function ,Reaction–diffusion system ,symbols ,Finite time ,Mathematical physics ,Mathematics - Abstract
This paper mainly considers the coupled parabolic system in a bounded domain: ut = Δu + uαvp, vt = Δv + uqvβ in Ω × (0, T) with null Dirichlet boundary value condition which had been discussed by Wang in (Z Angew Math Phys 51:160–167, 2000). The aim of this paper is to solve the open problem mentioned in the Remark of Wang (Z Angew Math Phys 51:160–167, 2000).
- Published
- 2013
32. Subsonic irrotational flows in a two-dimensional finitely long curved nozzle
- Author
-
Shangkun Weng
- Subjects
Applied Mathematics ,General Mathematics ,Nozzle ,Flow angle ,General Physics and Astronomy ,Geometry ,Mechanics ,Conservative vector field ,Flow (mathematics) ,Uniqueness ,Boundary value problem ,Subsonic and transonic wind tunnel ,Angle of inclination ,Mathematics - Abstract
This is a continuation of our previous paper Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf), where we have characterized a set of physical boundary conditions that ensures the existence and uniqueness of subsonic irrotational flow in a flat nozzle. In this paper, we will investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angle of inclination of nozzle walls play the same role as the end pressure for the stabilization of subsonic flows. In other words, the L2 and L∞ bounds of the derivative of these two quantities cannot be too large, similar as we have indicated in Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf) for the end pressure. The curvatures of the nozzle walls will also play an important role in the stability of the subsonic flow.
- Published
- 2013
33. Closed-form solution for Eshelby’s elliptic inclusion in antiplane elasticity using complex variable
- Author
-
Y. Z. Chen
- Subjects
Continuation ,Unit circle ,Applied Mathematics ,General Mathematics ,Linear form ,Mathematical analysis ,General Physics and Astronomy ,Conformal map ,Eigenstrain ,Elasticity (economics) ,Closed-form expression ,Physical plane ,Mathematics - Abstract
This paper provides a closed-form solution for the Eshelby’s elliptic inclusion in antiplane elasticity. In the formulation, the prescribed eigenstarins are not only for the uniform distribution, but also for the linear form. After using the complex variable and the conformal mapping, the continuation condition for the traction and displacement along the interface in the physical plane can be reduced to a condition along the unit circle. The relevant complex potentials defined in the inclusion and the matrix can be separated from the continuation conditions of the traction and displacement along the interface. The expressions of the real strains and stresses in the inclusion from the assumed eigenstrains are presented. Results for the case of linear distribution of eigenstrain are first obtained in the paper.
- Published
- 2013
34. Stabilization of a thermoelastic Mindlin–Timoshenko plate model revisited
- Author
-
Marié Grobbelaar-Van Dalsen
- Subjects
Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Geometry ,Thermal conduction ,Displacement (vector) ,symbols.namesake ,Thermoelastic damping ,Heat flux ,Dirichlet boundary condition ,Second sound ,symbols ,Boundary value problem ,Mathematics - Abstract
This paper is a continuation of our work in Grobbelaar-Van Dalsen (Appl Anal 90:1419–1449, 2011) where we showed the strong stability of models involving the thermoelastic Mindlin–Timoshenko plate equations with second sound. For the case of a plate configuration consisting of a single plate, this was accomplished in radially symmetric domains without applying any mechanical damping mechanism. Further to this result, we establish in this paper the non-exponential stability of the model for a particular configuration under mixed boundary conditions on the shear angle variables and Dirichlet boundary conditions on the displacement and thermal variables when the heat flux is described by Fourier’s law of heat conduction. We also determine the rate of polynomial decay of weak solutions of the model in a radially symmetric region under Dirichlet boundary conditions on the displacement and thermal variables and free boundary conditions on the shear angle variables.
- Published
- 2012
35. Pullback attractors of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows
- Author
-
Bo You and Fang Li
- Subjects
Forcing (recursion theory) ,Field (physics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Order (ring theory) ,Pullback attractor ,01 natural sciences ,Fractal dimension ,Upper and lower bounds ,010101 applied mathematics ,Liquid crystal ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
This paper is concerned with the long-time behaviour of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows introduced in Lin and Liu (Commun Pure Appl Math, 48:501–537, 1995) with a non-autonomous forcing bulk term and order parameter field boundary conditions. In this paper, we prove the existence of pullback attractors and estimate the upper bound of its fractal dimension under some suitable assumptions.
- Published
- 2016
36. Global smooth flows for compressible Navier–Stokes–Maxwell equations
- Author
-
Jiang Xu and Hongmei Cao
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Maxwell's equations ,Compressibility ,symbols ,Dissipative system ,Initial value problem ,Navier stokes ,0101 mathematics ,Mathematics - Abstract
Umeda et al. (Jpn J Appl Math 1:435–457, 1984) considered a rather general class of symmetric hyperbolic–parabolic systems: $$A^{0}z_{t}+\sum_{j=1}^{n}A^{j}z_{x_{j}}+Lz=\sum_{j,k=1}^{n}B^{jk}z_{x_{j}x_{k}}$$ and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices $${L}$$ and $${B^{jk}(j,k=1,\ldots,n)}$$ are both assumed to be real symmetric. So far there are no general results in case that $${L}$$ and $${B^{jk}}$$ are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of $${L^{1}({\mathbb{R}}^3)}$$ - $${L^2({\mathbb{R}}^3)}$$ type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of $${L^p({\mathbb{R}}^n)}$$ - $${L^{q}({\mathbb{R}}^n)}$$ - $${L^{r}({\mathbb{R}}^n)}$$ estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.
- Published
- 2016
37. Nonlinear, finite deformation, finite element analysis
- Author
-
Anthony M. Waas and Nhung Nguyen
- Subjects
Deformation (mechanics) ,Cauchy stress tensor ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Constitutive equation ,General Physics and Astronomy ,02 engineering and technology ,021001 nanoscience & nanotechnology ,Stress (mechanics) ,symbols.namesake ,Cauchy elastic material ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Rate of convergence ,Finite strain theory ,Jacobian matrix and determinant ,symbols ,0210 nano-technology ,Mathematics - Abstract
The roles of the consistent Jacobian matrix and the material tangent moduli, which are used in nonlinear incremental finite deformation mechanics problems solved using the finite element method, are emphasized in this paper, and demonstrated using the commercial software ABAQUS standard. In doing so, the necessity for correctly employing user material subroutines to solve nonlinear problems involving large deformation and/or large rotation is clarified. Starting with the rate form of the principle of virtual work, the derivations of the material tangent moduli, the consistent Jacobian matrix, the stress/strain measures, and the objective stress rates are discussed and clarified. The difference between the consistent Jacobian matrix (which, in the ABAQUS UMAT user material subroutine is referred to as DDSDDE) and the material tangent moduli (Ce) needed for the stress update is pointed out and emphasized in this paper. While the former is derived based on the Jaumann rate of the Kirchhoff stress, the latter is derived using the Jaumann rate of the Cauchy stress. Understanding the difference between these two objective stress rates is crucial for correctly implementing a constitutive model, especially a rate form constitutive relation, and for ensuring fast convergence. Specifically, the implementation requires the stresses to be updated correctly. For this, the strains must be computed directly from the deformation gradient and corresponding strain measure (for a total form model). Alternatively, the material tangent moduli derived from the corresponding Jaumann rate of the Cauchy stress of the constitutive relation (for a rate form model) should be used. Given that this requirement is satisfied, the consistent Jacobian matrix only influences the rate of convergence. Its derivation should be based on the Jaumann rate of the Kirchhoff stress to ensure fast convergence; however, the use of a different objective stress rate may also be possible. The error associated with energy conservation and work-conjugacy due to the use of the Jaumann objective stress rate in ABAQUS nonlinear incremental analysis is viewed as a consequence of the implementation of a constitutive model that violates these requirements.
- Published
- 2016
38. Asymptotic behavior of boundary blow-up solutions to elliptic equations
- Author
-
Shuibo Huang
- Subjects
Mean curvature ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Degenerate energy levels ,General Physics and Astronomy ,Boundary (topology) ,Function (mathematics) ,Infinity ,01 natural sciences ,Omega ,010101 applied mathematics ,Combinatorics ,Asymptotic formula ,0101 mathematics ,Power function ,Mathematics ,media_common - Abstract
This paper is concerned with the asymptotic behavior on $${\partial\Omega}$$ of boundary blow-up solutions to semilinear elliptic equations $$\left\{\begin{array}{ll} \Delta u=b(x)f(u),~~ &x\in \Omega, \\ u(x)=\infty, ~~ &x\in\partial\Omega,\end{array} \right.$$ where b(x) is a nonnegative function on $${\Omega}$$ and may vanish on $${\partial\Omega}$$ at a very degenerate rate; f is nonnegative function on [0,∞) and normalized regularly varying or rapidly varying at infinity. The main feature of this paper is to establish a unified and explicit asymptotic formula when the function f is normalized regularly varying or grows faster than any power function at infinity. The effect of the mean curvature of the nearest point on the boundary in the second-order approximation of the boundary blow-up solution is also discussed. Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.
- Published
- 2016
39. Time-periodic solutions of the compressible Navier–Stokes equations in $${\mathbb{R}^{4}}$$ R 4
- Author
-
Chunhua Jin
- Subjects
Time periodic ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Space dimension ,General Physics and Astronomy ,01 natural sciences ,010101 applied mathematics ,Fixed-point iteration ,Compressibility ,Uniqueness ,0101 mathematics ,Compressible navier stokes equations ,Mathematics - Abstract
This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).
- Published
- 2016
40. How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach 'à la D’Alembert'
- Author
-
Francesco dell’Isola, Angela Madeo, and Pierre Seppecher
- Subjects
business.product_category ,Deformation (mechanics) ,Continuum mechanics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Order (ring theory) ,Cauchy distribution ,Edge (geometry) ,Wedge (mechanical device) ,Tensor ,Representation (mathematics) ,business ,Mathematics - Abstract
Navier–Cauchy format for Continuum Mechanics is based on the concept of contact interaction between sub-bodies of a given continuous body. In this paper, it is shown how—by means of the Principle of Virtual Powers—it is possible to generalize Cauchy representation formulas for contact interactions to the case of Nth gradient continua, that is, continua in which the deformation energy depends on the deformation Green–Saint-Venant tensor and all its N − 1 order gradients. In particular, in this paper, the explicit representation formulas to be used in Nth gradient continua to determine contact interactions as functions of the shape of Cauchy cuts are derived. It is therefore shown that (i) these interactions must include edge (i.e., concentrated on curves) and wedge (i.e., concentrated on points) interactions, and (ii) these interactions cannot reduce simply to forces: indeed, the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green, and Germain) is fundamental and unavoidable in the theory of Nth gradient continua.
- Published
- 2012
41. Infinitely many solutions for a differential inclusion problem in $${\mathbb{R}^N}$$ involving p(x)-Laplacian and oscillatory terms
- Author
-
Bin Ge, Xiaoping Xue, and Qing-Mei Zhou
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,Zero (complex analysis) ,General Physics and Astronomy ,Function (mathematics) ,Type (model theory) ,Lipschitz continuity ,Variational method ,Differential inclusion ,Embedding ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the differential inclusion in $${\mathbb{R}^N}$$ involving the p(x)-Laplacian of the type $${\begin{array}{lll}-\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u\in \partial F(x,u(x)),\;\;{\rm in}\;\;\mathbb{R}^N,\quad\quad\quad\quad\quad\quad ({\rm P})\end{array}}$$ where $${p: \mathbb{R}^N \to {\mathbb{R}}}$$ is Lipschitz continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. Under suitable oscillatory assumptions on the potential F at zero or at infinity, we show the existence of infinitely many solutions of (P). We also establish a Bartsch-Wang type compact embedding theorem for variable exponent spaces.
- Published
- 2012
42. A doubly degenerate diffusion system multi-coupled via inner and boundary sources
- Author
-
Sining Zheng and Zhaoxin Jiang
- Subjects
Degenerate diffusion ,Parabolic system ,Nonlinear parameters ,Applied Mathematics ,General Mathematics ,Degenerate energy levels ,Mathematical analysis ,Energy method ,General Physics and Astronomy ,Boundary (topology) ,Cover (algebra) ,Blowing up ,Mathematics - Abstract
This paper deals with a doubly degenerate parabolic system multi-coupled by inner and boundary sources. The necessary-sufficient conditions for global weak solutions are determined, which involve a complete classification for all the eight nonlinear parameters of the model and cover all possible blowing up mechanisms of solutions. The results of the paper are mainly rely on the comparison principle and the energy method.
- Published
- 2011
43. Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity
- Author
-
Wenjie Gao and Bin Guo
- Subjects
Nonlinear system ,Elliptic partial differential equation ,Variable exponent ,Applied Mathematics ,General Mathematics ,Boundary problem ,Mathematical analysis ,General Physics and Astronomy ,Fixed-point theorem ,Parabolic cylinder function ,Uniqueness ,Parabolic partial differential equation ,Mathematics - Abstract
The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.
- Published
- 2011
44. Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability
- Author
-
Shi-Liang Wu, Hai-Qin Zhao, and San-Yang Liu
- Subjects
education.field_of_study ,Applied Mathematics ,General Mathematics ,Population ,Mathematical analysis ,General Physics and Astronomy ,Perturbation (astronomy) ,Monotone polygon ,Exponential stability ,Norm (mathematics) ,Traveling wave ,Uniqueness ,education ,Delayed reaction ,Mathematics - Abstract
This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossing-monostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not be monotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weighted energy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson’s blowflies equation in population dynamics and Mackey–Glass model in physiology.
- Published
- 2010
45. A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux
- Author
-
Jay I. Frankel, Jayne Wu, Rao V. Arimilli, and Majid Keyhani
- Subjects
Surface (mathematics) ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Mechanics ,Half-space ,Noise (electronics) ,Heat flux ,Control theory ,Development (differential geometry) ,Boundary value problem ,Transient (oscillation) ,Heat kernel ,Mathematics - Abstract
This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.
- Published
- 2007
46. Large time behavior for the non-Newtonian flow in R3
- Author
-
Luo Wu, Zongyao Wang, and Huizhao Liu
- Subjects
Physics::General Physics ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,General Physics and Astronomy ,Infinity ,Space (mathematics) ,Non newtonian flow ,Flow (mathematics) ,Control theory ,High Energy Physics::Experiment ,Mathematics ,media_common - Abstract
This paper interests a system for the non-Newtonian flow in the whole space. [14] estimated decay of it as t tends to infinity. The aim of the paper is to investigate decay problem of it and to improve a result of [14].
- Published
- 2007
47. Global existence and uniqueness of solution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum
- Author
-
Li Yin, Hongjun Yuan, and Xiaojing Xu
- Subjects
Class (set theory) ,Applied Mathematics ,General Mathematics ,Computation ,Mathematical analysis ,General Physics and Astronomy ,Interval (mathematics) ,Non-Newtonian fluid ,Physics::Fluid Dynamics ,General Relativity and Quantum Cosmology ,Bounded function ,Point (geometry) ,Uniqueness ,Boundary value problem ,Mathematics - Abstract
The aims of this paper are to discuss global existence and uniqueness of solution for a class of non-Newtonian fluids with vacuum in one-dimensional bounded interval. The important point in this paper is that we allow the initial vacuum. In particular, these results are used to prove similar results for more general non-Newtonian fluids, and applied to numerical computation.
- Published
- 2007
48. Explicit relation between the solutions of the heat and the Hermite heat equation
- Author
-
Bang-He Li
- Subjects
Discrete mathematics ,Pure mathematics ,Hermite polynomials ,Uniqueness theorem for Poisson's equation ,Applied Mathematics ,General Mathematics ,Bounded function ,Mathematics::Classical Analysis and ODEs ,General Physics and Astronomy ,Heat equation ,Constant (mathematics) ,Mathematics - Abstract
There are lots of results on the solutions of the heat equation $$\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u,$$ but much less on those of the Hermite heat equation $$\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U$$ due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).
- Published
- 2007
49. Continuous dependence of heat flux on spatial geometry for the generalized Maxwell-Cattaneo system
- Author
-
L. E. Payne and Changhao Lin
- Subjects
Partial differential equation ,Heat flux ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Spatial geometry ,Mathematics - Abstract
In the Generalized Maxwell-Cattaneo equations the temperature and heat flux are separate variables that are related through a system of partial differential equations. In a previous paper [5] the authors established continuous dependence of the temperature on spatial geometry. In this paper inequalities are derived which imply continuous dependence of the heat flux on spatial geometry. The arguments employed here are quite different and more complicated than those of the previous paper.
- Published
- 2004
50. [Untitled]
- Author
-
B. Cheng, David H. Sharp, and James Glimm
- Subjects
Entrainment (hydrodynamics) ,Turbulence ,Applied Mathematics ,General Mathematics ,Multiphase flow ,General Physics and Astronomy ,Mechanics ,Physics::Fluid Dynamics ,Constraint (information theory) ,Closure (computer programming) ,Volume (thermodynamics) ,Compressibility ,Statistical physics ,Diffusion (business) ,Mathematics - Abstract
In this paper we formulate a multiphase model with nonequilibrated temperatures but with equal velocities and pressures for each species. Turbulent mixing is driven by diffusion in these equations. The closure equations are defined in part by reference to a more exact chunk mix model developed by the authors and coworkers which has separate pressures, temperatures, and velocities for each species. There are two main results in this paper. The first is to identify a thermodynamic constraint, in the form of a process dependence, for pressure equilibrated models. The second is to determine one of the diffusion coefficients needed for the closure of the equilibrated pressure multiphase flow equations, in the incompressible case. The diffusion coefficients depend on entrainment times derived from the chunk mix model. These entrainment times are determined here first via general formulas and then explicitly for Rayleigh-Taylor and Richtmyer-Meshkov large time asymptotic flows. We also determine volume fractions for these flows, using the chunk mix model.
- Published
- 2002
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