Search

In this paper, we pursue our series of papers aiming to show the applicability of the concept of very weak solutions. We consider a wave model with irregular position-dependent mass and dissipation terms, in particular, allowing for δ -like coefficients and prove that the problem has a very weak solution. Furthermore, we prove the uniqueness in an appropriate sense and the coherence of the very weak solution concept with classical theory. A special case of the model considered here is the so-called telegraph equation. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for a two-component peakon system with cubic nonlinearity, which is a natural multi-component extension of the single Camassa-Holm equation. We first establish the local well-posedness for the Cauchy problem of the system, and then derive a blow-up criteria and construct data that lead to the finite time blow-up by exploiting a special conservation law and by using the method of characteristics. It is worthwhile to point out that the classical approach to study the blow up phenomena heavily depends on the control of H 1 -norm of the velocity component. However, this two-component peakon system considered in this paper does not admit H 1 -norm conservation law. Our idea is to use the new conservation law H 1 = 1 6 ∫ R (u - u x) n d x = 1 6 ∫ R (v + v x) m d x (see Lemma 3.2) and the structure of the system to obtain the estimates on ‖ m ‖ L 1 and ‖ n ‖ L 1 (see Lemma 3.3). As a result, we can assert that the finite time blow-up can occur if and only if the slope of the transport velocity is unbounded below and derive a new blow-up result for strong solutions to the system. Finally, we also establish the persistence properties of the solutions to the integrable peakon system in weighted L ϕ p spaces for a large class of moderate weights. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the nonlinear Schrodinger system {idtu1(x,t)== ∆u1(x, t)− |u1(x, t)| p−1u1(x, t)− |u2(x, t)| p−1u1(x, t), idtu2(x, t) = ∆u2(x, t) − |u2(x, t)| p−1u2(x, t) − |u1(x, t)| p−1u2(x, t), was investigated in d space dimensions. For 1 < p ≤ 1 + 2/d, the nonexistence of asymptotically free solutions for the nonlinear Schrödinger system was proved based on mathematical analysis and scattering theory methods. The novelty of this paper was to give the proof of pseudo-conformal identity on the nonlinear Schrödinger system. The present results improved and complemented these of Bisognin, Sepúlveda, and Vera(Appl. Numer. Math. 59(9)(2009): 2285–2302), in which they only proved the nonexistence of asymptotically free solutions when d = 1, p = 3. [ABSTRACT FROM AUTHOR]

The paper examines stochastic diffusion within an expanding space–time framework motivated by cosmological applications. Contrary to other results in the literature, for the considered general stochastic model, the expansion of space–time leads to a class of stochastic equations with non-constant coefficients that evolve with the expansion factor. The Cauchy problem with random initial conditions is posed and investigated. The exact solution to a stochastic diffusion equation on the expanding sphere is derived. Various probabilistic properties of the solution are studied, including its dependence structure, evolution of the angular power spectrum and local properties of the solution and its approximations by finite truncations. The paper also characterizes the extremal behaviour of the random solution by establishing upper bounds on the probabilities of large deviations. Numerical studies are carried out to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we studied the stability of the Cauchy problem for a class of first-order linear quasi-periodically forced PDEs on the m-dimensional torus: {∂tu + (ξ + f(x, ωt, ξ)) · ∂xu = 0, u(x, 0) = u0(x), where ξ ∈ ℝm, x ∈ Tm, ω ∈ ℝd, in the case of multidimensional Liouvillean forced frequency. We proved that for each compact set O ∈ ℝm there exists a Cantor subset Oγ of O with positive Lebesgue measure such that if ξ ∈ Oγ, then for a perturbation f being small in some analytic Sobolev norm, there exists a bounded and invertible quasi-periodic family of linear operator Ψ(ωt), such that the above PDEs are reduced by the transformation v := Ψ(ωt)−1 [u] into the following PDE: ∂tv + (ξ + m∞(ωt)) · ∂xv = 0, provided that the forced frequency ω ∈ ℝd possesses finite uniform Diophantine exponent, which allows Liouvillean frequency. The reducibility can immediately cause the stability of the above Cauchy problem, that is, the analytic Sobolev norms of the Cauchy problem are controlled uniformly in time. The proof is based on a finite dimensional Kolmogorov-Arnold-Moser (KAM) theory for quasi-periodically forced linear vector fields with multidimensional Liouvillean forced frequency. As we know, the results on Liouvillean frequency existing in the literature deal with two-dimensional frequency and exploit the theory of continued fractions to control the small divisor problem. The results in this paper partially extend the analysis to higher-dimensional frequency and impose a weak nonresonance condition, i.e., the forced frequency ω possesses finite uniform Diophantine exponent. Our result can be regarded as a generalization of analytic cases in the work [R. Feola, F. Giuliani, R. Montalto and M, Procesi, Reducibility of first order linear operators on tori via Moser’s theorem, J. Funct. Anal., 2019] from Diophantine frequency to Liouvillean frequency. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space Lp(.). The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the Lp(.) is transformed into the classical Lebesgue spaces. Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties. [ABSTRACT FROM AUTHOR]

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when α → 1 − . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when α → 1 − . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the Camassa–Holm equation in Sobolev spaces. Firstly, we establish the energy conservation for weak solutions of the Camassa–Holm equation in H 1 (R) ∩ B 3 , 2 1 (R) and prove that every weak solution in H 7 6 (R) is unique by the embedding H 7 6 (R) ↪ B 3 , 2 1 (R). Then, we obtain the local well-posedness for the Camassa–Holm equation in W 2 , 1 (R). It is worth noting that B 1 , 1 2 (R) ↪ W 2 , 1 (R) and the Camassa–Holm equation is well-posed in B 1 , 1 2 (R) and is ill-posed in W 1 + 1 p , p (R) (1 < p ≤ ∞) by the pervious papers. Our result implies that W 2 , 1 (R) is the critical Sobolev spaces for the well-posedness of the Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form L 1 + L 2 − ∂ t , but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators L 1 − ∂ t and L 2 − ∂ t . Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander's 1967 groundbreaking paper on hypoellipticity. [ABSTRACT FROM AUTHOR]

In this paper, the initial value problem for the third-order linear differential equation with small parameters at the two highest derivatives and piecewise-constant argument was considered when the roots of additional characteristic equation have negative signs. The aim of this paper is to obtain an explicit formula and the asymptotic estimations of the solution. The fundamental system of solutions, initial functions are constructed and their asymptotic estimations are obtained. It is also shown that the solution of the given problem at the points , have the phenomenon of an initial jump of the first order of the second degree. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel system with a fractional dissipation and advection by a weakly mixing flow (see Definition 2.5). Here the attractive kernel has a strong singularity, namely, the derivative appearing in the nonlinear term by singular integral. Without advection, the solution of system blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case that the order of the derivative of dissipative term is higher than that of nonlinear term. Since the attractive kernel has a strong singularity, the weakly mixing has both destabilizing effects and enhanced dissipation effects, which makes the problem more complicated and difficult. In this paper, we establish the $ L^\infty $-criterion and obtain the global $ L^\infty $ estimate of the solution by introducing some new ideas and techniques. [ABSTRACT FROM AUTHOR]

In this paper, the existence and the uniqueness of an evolutionary anisotropic $ p_i(x) $ p i (x) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the $ L^{\infty } $ L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the $ L^{\infty } $ L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic $ p(x)- $ p (x) − Laplace equation with convection term via $ L^{\infty }- $ L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for a class of coupled system of the four-dimensional cubic focusing nonlinear Schrödinger equations was investigated. By exploiting the double Duhamel method and the long-time Strichartz estimate, the global well-posedness and scattering were proven for the system below the ground state. In our proof, we first established the variational characterization of the ground state, and obtained the threshold of the global well-posedness and scattering. Second, we showed that the non-scattering is equivalent to the existence of an almost periodic solution by following the concentration-compactness/rigidity arguments of Kenig and Merle [17] (Invent. Math., 166 (2006), 645–675). Then, we obtained the global well-posedness and scattering below the threshold by excluding the almost periodic solution. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented. [ABSTRACT FROM AUTHOR]

We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with critical nonlinearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its optimal and sharp convergence rate of solutions with a critical exponent and two exponentially approaching initial data. This rate contains a logarithmic term which does not contain in the super critical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for a system of nonlinear Schrödinger equations with quadratic interactions and initial data belonging to a class of analytic Gevrey functions. Here, we present a local well-posedness result in the analytic Gevrey class G σ , s × G σ , s by proving some bilinear estimates in Bourgain's space with exponential weight. Furthermore, we prove that the obtained solution can be extended to any time T > 0 , as long as the radius of the spatial analyticity σ is bounded below by c T - 2 , if 0 < a < 1 / 2 , or c T - 4 , if a > 1 / 2 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem of Keller–Segel–Navier–Stokes system with fractional diffusion. Making use of Fourier localization technique and Littlewood-Paley theory, we establish the global well-posedness of mild solution for small initial data in mixed time-space Besov spaces. Furthermore, we obtain the well-posedness of mild solution in time-weighted Besov spaces by Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

We investigate the space-time decay rates of solutions to the 3D Cauchy problem of the compressible Oldroyd-B system with diffusive properties and without viscous dissipation. The main novelties of this paper involve two aspects: On the one hand, we prove that the weighted rate of k-th order spatial derivative (where 0≤k≤3) of the global solution (ρ,u,η,τ) is t-3/4+k/2+γ in the weighted Lebesgue space L²γ. On the other hand, we show that the space-time decay rate of the m-th order spatial derivative (where m∈[0,2]) of the extra stress tensor of the field in L02γ is (1+t)-5/4-m/2+γ, which is faster than that00 of the velocity. The proofs are based on delicate weighted energy methods and interpolation tricks. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the maximum norm of velocity divergence controls the breakdown of smooth (strong) solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier–Stokes equations with zero heat conduction. The results indicate that the nature of the blowup for the full compressible Navier–Stokes equations with zero heat conduction of viscous flow is similar to the barotropic compressible Navier–Stokes equations and does not depend on the temperature field. The main ingredient of the proof is a priori estimate to the pressure field instead of the temperature field and weighted energy estimates under the assumption that velocity divergence remains bounded. Furthermore, the initial vacuum states are allowed, and the viscosity coefficients are only restricted by the physical conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to new propagation and regularity results for a class of first-order Hamilton–Jacobi–Bellman-type problems in a separable infinite-dimensional Hilbert space. Specifically, the related Cauchy problem is investigated by employing the Faedo–Galerkin approximation method. Under some structural assumptions, the main result is obtained by using the probabilistic representation formula of the solution in order to define the weak continuity assumptions, by assuming the existence of a symmetric positive definite Hilbert–Schmidt operator and by employing modulus continuity arguments. [ABSTRACT FROM AUTHOR]

The Ablowitz–Ladik equations, hereafter called A L + and A L − , are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schrödinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of A L ± in the periodic setting, showing in particular that the background solution of A L − is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than 1, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of A L ± describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of A L − are always singular on curves of spacetime, if they live on a background of sufficiently large amplitude, and we construct a different continuous limit describing this regime: a NLS equation with a nonlinear and weak dispersion. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution (i) evolves according to A L + into a recurrence of the above exact solutions, in the case of one and two unstable modes, and (ii) evolves according to A L − into a singularity in finite time if the amplitude of the background is greater than 1. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect. [ABSTRACT FROM AUTHOR]

Consider the second order nonlinear partial differential equation:... Given small analytic data, Yamane was able to obtain the order of the lifespan of the solution with respect to the smallness parameter e. On the other hand, Gourdin and Mechab studied the lifespan of the solution given small Gevrey data, but under the assumption that F is independent of u. In this paper, we considered non-vanishing Gevrey data and used the method of successive approximations to obtain a solution and constructively estimate its lifespan. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for Cahn–Hilliard equation in R n ( n ∈ Z + , n ≥ 3 ). First, based on the higher-order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Then, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H n 2 + 1 -norm of initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we are dealing with the ill-posed Cauchy problem for an elliptic operator. This is a follow-up to a previous paper on the same subject. Indeed, in an earlier publication, we introduced a regularization method, called the controllability method, which allowed us to propose, on the one hand, a characterization of the existence of a regular solution to the ill-posed Cauchy problem. On the other hand, we have also succeeded in proposing, via a strong singular optimality system, a characterization of the optimal solution to the considered control problem, and this, without resorting to the Slater-type assumption, an assumption to which many analyses had to resort. On occasion, we have dealt with the control problem, with state boundary observation, the problem initially analyzed by J. L. Lions. The proposed point of view, consisting of the interpretation of the Cauchy system as a system of two inverse problems, then called naturally for conjectures in favor of which the present manuscript wants to constitute an argument. Indeed, we conjectured, in view of the first results obtained, that the proposed method could be improved from the point of view of the initial interpretation that we had made of the problem. In this sense, we analyze here two other variants (observation of the flow, then distributed observation) of the problem, the results of which confirm the intuition announced in the previous publication mentioned above. Those results, it seems to us, are of significant relevance in the analysis of the controllability method previously introduced. [ABSTRACT FROM AUTHOR]

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al. 2015 Solutions to an advanced functional partial differential equation of the pantograph type (Proc. R. Soc. A471, 20140947 (doi:10.1098/rspa.2014.0947)). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we study the global wellposedness of a radiation hydrodynamics model with viscosity and thermal conductivity. It is now well-understood that, unlike the compressible Euler equations whose smooth solutions must blow up in finite time no matter how small and how smooth the initial data is, the dissipative structure of such a radiation hydrodynamics model can indeed guarantee that its one-dimensional Cauchy problem admits a unique global smooth solution provided that the initial data is sufficiently small, while for large initial data, even if the heat conductivity is taken into account but the viscosity effect is ignored, shock type singularities must appear in finite time for smooth solutions of the Cauchy problem of a one-dimensional radiation hydrodynamics model with thermal conductivity and zero viscosity. Thus a natural question is, If effects of both the viscosity and the thermal conductivity are considered, does the one-dimensional radiation hydrodynamics model with viscosity and thermal conductivity exist as a unique global large solution? We give an affirmative answer to this problem and show in this paper that the initialboundary value problem to the radiation hydrodynamics model in a one-dimensional periodic box T = R/Z with viscosity and thermal conductivity does exist as a unique global smooth solution for any large initial data. The main ingredient in our analysis is to introduce some delicate estimates, especially an improved Lm ([0, T], Lco(T))-estimate on the absolute temperature for some m E N and a pointwise estimate between the absolute temperature, the specific volume, and the first-order spatial derivative of the macro radiation flux, to deduce the desired positive lower and upper bounds on the density and the absolute temperature. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of global entropy solutions for the Cauchy problem of an isentropic gas dynamics system with the special pressure P(r) = 1/1-ρ. After the gas density ρ is fixed in the region ρ ε (0,1), by the method of the artificial viscosity and the maximum principle, this system is nonstrictly hyperbolic and genuinely nonlinear, and its global entropy solutions are obtained by the famous compactness framework introduced by DiPerna in the paper "Convergence of approximate solutions to conservation laws" (Arch. Rat. Mech. Anal., (82) (1983), 27-70). [ABSTRACT FROM AUTHOR]

It is known that the damped fractional wave equation has the diffusive structure as t → ∞ . Let u (t , x) = e − t cosh (t L) f (x) + e − t sinh (t L) L (f (x) + g (x)) be the solution of the Cauchy problem for the damped fractional wave equation, where L involves the fractional Laplacian (− △) α on the space variable. We can study the decay estimate of the solution u (t , x) over the time t by means of the Cauchy problem for the parabolic equation. In this paper, we consider, for 0 < α < 1 , the Cauchy problem in the two- and three-dimensional spaces for the damped fractional wave equation and the corresponding parabolic equation and obtain the Triebel–Lizorkin space estimate of the difference of solutions. At the same time, we also consider, for α = 1 , the case of the Cauchy problem in the four-dimensional space and obtain a Triebel–Lizorkin space estimate. [ABSTRACT FROM AUTHOR]

This paper aims to show that the Cauchy problem of the Burgers equation with a weakly dispersive perturbation involving the Bessel potential (generalization of the Fornberg–Whitham equation) can exhibit wave breaking for initial data with large slope. We also comment on the dispersive properties of the equation. [ABSTRACT FROM AUTHOR]

In this paper, we study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Lefflertype function to study oscillatory-type integrals appearing in the analysis of time-fractional partial differential equations. Several generalisations of the first and second van der Corput lemmas are proved. Optimal estimates on decay orders for particular cases of the Mittag-Leffler functions are also obtained. As an application of the above results, the generalised Riemann–Lebesgue lemma and the Cauchy problem for the time-fractional evolution equation are considered. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence and uniqueness of solution for a large class of perturbed sweeping processes formulated by fractional differential inclusions in infinite dimensional setting. The normal cone to the (mildly non-convex) prox-regular moving set C(t) is supposed to have a Hölder continuous variation, is perturbed by a continuous mapping, which is both time and state dependent. Using an explicit catching-up algorithm, we show that the fractional perturbed sweeping process has one and only one Hölder continuous solution. Then this abstract result is applied to provide a theorem on the weak solvability of a fractional viscoelastic frictionless contact problem. The process is quasistatic and the constitutive relation is modeled with the fractional Kelvin–Voigt law. This application represents an additional novelty of our paper. [ABSTRACT FROM AUTHOR]

Copyright of Journal of University of Anbar for Pure Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider the Cauchy problem for (abcd)-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona et al. (J Nonlinear Sci 12:283–318, 2002, Nonlinearity 17:925–952, 2004), describes a small-amplitude waves on the surface of an inviscid fluid, and is derived as a first order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of one-dimensional BBM–BBM case by Chen and Liu (Anal Math 121:299–316, 2013). Among results established here, we emphasize that the ill-posedness result for two-dimensional BBM–BBM system is optimal. The proof follows from an observation of the high to low frequency cascade present in nonlinearity, motivated by Bejenaru and Tao (J Funct Anal 233:228–259, 2006). [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of trajectories of systems of ordinary differential equations generated by the velocity field of a moving incompressible viscoelastic fluid with memory along the trajectories in a domain with multiple boundary components. The case of a velocity field from a Sobolev space with inhomogeneous boundary conditions is considered. The properties of the maximal intervals of existence of solutions to the Cauchy problem corresponding to a given velocity field are investigated. The study assumes the approximation of a velocity field by a sequence of smooth fields followed by a passage to the limit. The theory of regular Lagrangian flows is used. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterratype integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order d, 0 < d < 1. [ABSTRACT FROM AUTHOR]

This paper was concerned with the Cauchy problem of the 3D magnetohydrodynamic (MHD) system. We first proved that this system was local well-posed with initial data in the Besov space , in the critical Besov space , and in with , respectively. We also obtained a new growth rate estimates for the analyticity radius. [ABSTRACT FROM AUTHOR]

In this paper, we study the quantitative stability of an optimal control problem with respect to parametric perturbations. We essentially obtain two equivalent conclusions for the stability of this problem by using two independent methods. The first one makes recourse to standard computations based on the famous Gronwall Lemma while our second method employees rather stability of fixed points trough the celebrated Lim's Lemma for which we construct a suitable contracting set-valued mapping over a larger functional space than the one of continuous functions adopted in the close previous works. The second method allows us to introduce a further concept of approximate solutions regarded as approximate values of the optimal control for which we prove similar stability properties as in the case of exact solutions. [ABSTRACT FROM AUTHOR]

The main purpose of this investigation is to revisit solvability process of the Cauchy problems of Caputo fractional two-term initial value problems. To this aim, the Green function technique has chosen to make a bridge between the operator and the fixed point theories. The appeared Green functions in this paper are constructed by the Fox-Wright functions. Our solvability tools include the existence and uniqueness criteria as novel refinements of the Banach contraction principal and Schauder fixed point theorem. This investigation will be finalized by presenting some numerical applications that illustrate proposed solvability criteria. [ABSTRACT FROM AUTHOR]

The analysis of a complex system requires the development of suitable mathematical structures able to take into account the multi-agent role. The aim of this paper is the derivation of a preliminary multi-agent framework of the discrete thermostatted kinetic theory for active particles. According to the proposed framework the overal system is divided into primary active particle-subsystems which are subsequently grouped into different functional subsystems composed by active particles sharing the same internal state. The new framework consists into a system of nonlinear ordinary differential equations. The paper is addressed to the existence and uniqueness of the solution of the related Cauchy problem. The main result is obained by employing ODE arguments and Li estimations. The applications include, but are not limited, to vehicular traffic, crowd dynamics, swarm dynamics, biological, economic, social, and engineering systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity. [ABSTRACT FROM AUTHOR]

This paper deals with the Cauchy problem of 3D inhomogeneous incompressible magnetic Bénard equations. Through some time-weighted a priori estimates, we prove the global existence of strong solution provided that the upper boundedness of initial density and initial magnetic field satisfy some smallness condition. Furthermore, we also obtain large time decay rates of the solution. [ABSTRACT FROM AUTHOR]

The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps. [ABSTRACT FROM AUTHOR]

This paper is concerned with time-asymptotic nonlinear stability of rarefaction waves to the Cauchy problem for one-dimensional compressible non-isentropic magnetohydrodynamics (MHD) equations (including its isentropic case), which describe the motion of a conducting fluid in a magnetic field. Through some elaborate and rigorous mathematical analysis, we can construct the rarefaction waves v r , u r , θ r , b r (x / t) where magnetic component b r x / t is a nontrivial profile, namely a non-constant function. Then the solution of the compressible MHD equations is proved to tend towards the rarefaction waves time-asymptotically under small initial perturbations and weak wave strength, and also under a technical assumption that the parameter β = v + b + is bounded by a specific constant. The proof of the main result is based on elementary L 2 energy methods. [ABSTRACT FROM AUTHOR]