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We consider the volume-preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long-time asymptotics approach round spheres. The proofs are based on a priori estimates on the inner and outer radii of the solutions.

We consider a class of linear second order differential equations with damping and external force. We investigate the link between a uniform bound on the forcing term and the corresponding ultimate bound on the velocity of solutions, and we study the dependence of that bound on the damping and on the "elastic force". We prove three results. First of all, in a rather general setting we show that different notions of bound are actually equivalent. Then we compute the optimal constants in the scalar case. Finally, we extend the results of the scalar case to abstract dissipative wave-type equations in Hilbert spaces. In that setting we obtain rather sharp estimates that are quite different from the scalar case, in both finite and infinite dimensional frameworks. The abstract theory applies, in particular, to dissipative wave, plate and beam equations., Comment: 29 pages

Made available in DSpace on 2020-12-12T01:51:25Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-01-01 We study a plate equation with presumed nonuniqueness for the associated Cauchy problem. We establish the existence of global weak solutions by the Faedo–Galerkin method, and our main result refers to the existence of a global attractor using the method of generalized semiflows. Department of Physical Chemistry Institute of Chemistry UNESP Department of Physical Chemistry Institute of Chemistry UNESP

We study the asymptotic behavior of solutions of the differential equation of Emden-Fowler type y'' - xayσ = 0, where a _ R, σ > 1. We prove some of theorems on asymptotic properties of solutions and obtain asymptotic formulas for solutions near zero.

Accurate asymptotic formulas for regularly varying solutions of the second order half-linear differential equation \begin{equation*} (|x'|^{\alpha}\textrm{sgn}\; x')' + q(t)|x|^{\alpha}\textrm{sgn}\; x = 0, \end{equation*} will be established explicitly, depending on the rate of decay toward zero of the function \begin{equation*}Q_c(t) = t^{\alpha}\int_t^{\infty}q(s)ds - c\end{equation*} as $t\to\infty$, where $c

In this paper we investigate a logistic equation with a new free boundary condition appearing in ecology, we aim to describe the spreading of a new or invasive species by studying the asymptotic behavior of the radially symmetric solutions of the problem. We will obtain a trichotomy result: spreading (the solution converges to a stationary solution defined on the half–line), transition (the solution converges to a stationary solution with compact support) and vanishing (the solution converges to 0 within a finite time). Besides we can also obtain a dichotomy result (either spreading or vanishing happens). Moreover, in the spreading case, we give the sharp estimate of the asymptotic spreading speed of the free boundary.

Positive solutions of the nonlinear second-order differential equation $(p(t)|x'|^{\alpha - 1} x')' + q(t)|x|^{\beta - 1} x = 0,\alpha > \beta > 0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.

The asymptotic behavior of solutions of the odd-order differential equation of Emden-Fowler type $$ x^{(2n+1)}(t) + q(t)|x(t)|^{\gamma}\textrm{sgn}\;x(t)=0 , $$ is studied in the framework of regular variation, under the assumptions that $0

This paper is concerned with asymptotic analysis of positive solutions of the second-order nonlinear differential equation (A)x^'^'(t)+q(t)@f(x(t))=0, where q:[a,~)->(0,~) is a continuous function which is regularly varying and @f:(0,~)->(0,~) is a continuous increasing function which is regularly varying of index @c@?(0,1). An application of the theory of regular variation gives the possibility of determining precise information about the asymptotic behavior at infinity of intermediate solutions of Eq. (A).

The one-dimensional motion of compressible viscous and heat-conductive fluid is investigated in the half space. By examining the sign of fluid velocity prescribed on the boundary, initial boundary value problems with Dirichlet type boundary conditions are classified into three cases: impermeable wall problem, inflow problem and outflow problem. In this paper, the asymptotic stability of the rarefaction wave, boundary layer solution, and their combination is established for both the impermeable wall problem and the inflow problem under some smallness conditions. The proof is given by an elementary energy method.

We consider a transmission problem in thermoelasticity with memory. We show the exponential decay of the solution in case of radially symmetric situations, as time goes to infinity.

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i ( u i ) may have the property D i ( 0 ) = 0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.

In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated to this equation. We prove that the solution blows up at finite time $T$ if and only if its energy is negative at some time before $T$. The proof of this result is based on a Gamma-convergence technique.

The asymptotic behavior of the nonoscillatory solutions of quasilinear differential equations of second order with delay depending on the unknown function is considered. The main results given by [Bainov et al. (J. Comput. Appl. Math. 91 (1998) 87–96) and Wong (Funkcial. Ekvac. 11 (1968) 207–234)] are improved and generalized.

A class of second-order abstract systems with memory and Dirichlet boundary conditions is investigated. By suitable Liapunov functionals, existence of solutions as well as asymptotic behavior, are determined. In particular, when the memory kernel decays exponentially, the polynomially decay of the solutions is proved.

Some expressions for the free energy in the case of incompressible viscoelastic fluids are given. These are derived from free energies already introduced for other viscoelastic materials, adapted to incompressible fluids. A new free energy is given in terms of the minimal state descriptor. The internal dissipations related to these different functionals are also derived. Two equivalent expressions for the minimum free energy are given, one in terms of the history of strain and the other in terms of the minimal state variable. This latter quantity is also used to prove a theorem of existence and uniqueness of solutions to initial boundary value problems for incompressible fluids. Finally, the evolution of the system is described in terms of a strongly continuous semigroup of linear contraction operators on a suitable Hilbert space. Thus, a theorem of existence and uniqueness of solutions admitted by such an evolution problem is proved.

In this paper, we consider the quasilinear elliptic equation in a smooth bounded domain. By using the method of lower and upper solutions, we study the existence, asymptotic behavior near the boundary and uniqueness of the positive blow-up solutions for quasilinear elliptic equation with nonlinear gradient term. En este artículo consideramos la ecuación elíptica cuasilineal en un dominio acotado suave. Usando el método de sub y súper soluciones, estudiamos la existencia, comportamiento asintótico cerca de la frontera y la unicidad de soluciones explosivas para ecuaciones elípticas cuasilineales con término del gradiente nolineal.

We consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u tt +u t − div (a(∇u)∇u)=0, and show that they tend, as t→+∞ , to those of the nonlinear parabolic equation v t − div (a(∇v)∇v)=0, in the sense that the norm ‖u(. ,t)−v(. ,t)‖ L ∞ ( R n ) of the difference u−v decays faster than that of either u or v . This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by L. Hsiao and Tai-ping Liu.

In this paper, we introduce a new method for investigating the rate and profile of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blow-up and that the rate of blow-up is uniform in all compact subsets of the domain. This results in a flat blow-up profile, except for a boundary layer, whose thickness vanishes astapproaches the blow-up timeT*. In each case, the blow-up rate of |u(t)|∞is precisely determined. Furthermore, in many cases, we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer. The size of the boundary layer then decays like T *− t , and the solutionu(t, x) behaves like |u(t)|∞ d(x)/ T *− t in the boundary layer, wheredis the distance to the boundary. Some Fujita-type critical exponents results are also given for the Cauchy problem.

The blowup asymptotics for classical solutions of ut = Δu + eu + g(t) are analyzed, including the nonlocal case where g has a functional dependence of the form ∫ωutdx. The method of analyzing the evolution of the blowup singularity is a dynamical systems approach using center manifold techniques and analyzing an ω-limit set for a corresponding Hamilton-Jacobi equation.

The paper is devoted to the study of the solvability of a singular initial value problem for systems of ordinary differential equations. The main results give sufficient conditions for the existence of solutions in the right-hand neighbourhood of a singular point. In addition, the dimension of the set of initial data generating such solutions is estimated. An asymptotic behavior of solutions is determined as well and relevant asymptotic formulas are derived. Příspěvek je věnován studiu řešitelnosti singulární počáteční úlohy pro systémy obyčejných diferenciálních rovnic. Hlavními výsledky jsou postačující podmínky pro existenci řešení v pravém okolí singulárního bodu. Je stanovena i dimense množiny počátečních podmínek generujících taková řešení . Rovněž je popsáno i asymptotické chování řešení včetně stanovení asymptotických tvarů řešení.

The feature of the present work is to demonstrate that the method of regular variation can be effectively applied to fourth order quasilinear differential equations of the forms \begin{equation*} (|x''|^{\alpha-1}x'')'' + q(t)|x|^{\beta-1}x = 0, \end{equation*} under the assumptions that $\alpha \gt \beta$ and $q(t): [a,\infty) \to (0,\infty)$ is regularly varying function, providing full information about the existence and the precise asymptotic behavior of all possible positive solutions.

We study the asymptotic behavior of an incompressible viscoelastic fluid and prove that the temporal decay of the energy is similar to one of the memory kernel. The innovative aspect of this research lies in considering the evolutive problem with non-zero external sources and/or initial histories. Copyright © 2012 John Wiley & Sons, Ltd.

In this well-written paper, the authors consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. \par In the first part of the paper, some notions concerning dissipative systems in ordered space are recalled. Then follow results on the existence of extremal solutions and global attractors and finally on the inclusion of the global attractor in an order interval formed by the minimal and the maximal equilibria. \par In the second part of the paper, they then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, they exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in $\Bbb R^N$ with nonlinearities depending on the gradient of the solution. \par The authors consider as well systems of reaction-diffusion equations in $\Bbb R^N$ and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in $\Bbb R^N$. They further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation.

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{��/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for alpha in (1,2), the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for alpha \leq 1. If alpha=1, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case alpha \in (0,1), we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions., 23 pages. To appear to SIMA. This version contains details that are skipped in the published version

We study the existence and nonexistence of positive solutions to a sublinear ( p 1 ) second-order divergence type elliptic equation ( * ) : − ∇ ⋅ a ⋅ ∇ u = u p in unbounded cone-like domains C Ω . We prove the existence of the critical exponent p * ( a , C Ω ) = sup { p 1 : ( * ) has a positive supersolution at infinity in C Ω } , which depends on the geometry of the cone C Ω and the coefficients a of the equation.

International audience; We consider the solutions of the equation −ε2Δu+u−|u|p−1u=0 in S1×R, where ε and p are positive real numbers, p>1. We prove that the set of the positive bounded solutions even in x1 and x2, decreasing for x1∈]−π,0[ and tending to 0 as x2 tends to +∞ is the first branch of solutions constructed by bifurcation from the ground-state solution. We prove that there exists a positive real number ε⋆ such that for every ε∈]0,ε⋆] there exists a finite number of solutions verifying the above properties and none such solution for ε>ε⋆. The proves make use of compactness results and of the Leray-Schauder degree theory.

A parabolic equation in two or three space variables with a Preisach hysteresis operator and with homogeneous Neumann boundary conditions is shown to admit a unique global regular solution. A detailed investigation of the Preisach memory dynamics shows that the system converges to an equilibrium in the state space of all admissible Preisach memory configurations as time tends to infinity.

We study the asymptotic properties of solutions of the differential equation ?(t) = -c(t)[x(t) - Lx(τ(t))] with a positive continuous function c(t), a nonzero real constant L and unbounded lag. We establish conditions under which each solution of this equation approaches a solution of the auxiliary functional equation ψ(t) = Lψ(τ(t)). Moreover, we investigate some modifications of the studied equation and give comparisons with the known results.

金沢大学理工研究域電子情報学系, In the present paper, the dependencies of the numerical results of fluid simulations on forcibly added randomness are discussed. The incompressible Navier-Stokes equations and the continuity equation are solved numerically by using the MAC (Maker-And-Cell) method and implicit temporal scheme. The model adopted in the present study is a flow around a two-dimensional circular cylinder and the Reynolds number is 1500. The randomness which is given by using the pseudo-random number is forcibly added in the time marching step of the discretized Navier-Stokes equations. Dependencies of the averaged structure of asymptotic numerical solutions on the randomness are discussed. Furthermore, the dependence of the qualitative structure of the asymptotic solution of each sample calculation on the amplitude of randomness is also studied. It is clarified that forcibly added random errors may cover the nonlinear errors which make the system unstable. © 2008 Elsevier B.V. All rights reserved.

In this paper we exhibit the dissipative mechanism of the Cahn–Hilliard equation in H1(RN). We show a weak form of dissipativity by showing that each individual solution is attracted, in some sense, by the set of equilibria. We also indicate that strong dissipativity, that is, asymptotic compactness in H1(RN), cannot be in general expected. Then we consider two types of perturbations: a nonlinear perturbation and a small linear perturbation. In both cases we show that, for the resulting equations, the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor.

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered isX′=f(t,X)and the associated perturbed differential equation isY′=f(t,Y)+g(t,Y).The approach used is to examine the difference between the respective solutionsF(t,t0,x0)andG(t,t0,y0)of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the differenceG(t,t0,y0)−F(t,t0,x0)in the case where the initial valuesy0andx0are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.