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In this paper, we will show that solutions of the initial boundary value problem for the coupled system of Allen–Cahn/Cahn–Hilliard equations continuously depend on parameters of the system, and under some restrictions on the parameters all solutions of the initial boundary value problem for Allen–Cahn/Cahn–Hilliard equations tend to zero with an exponential rate as $t\rightarrow\infty$ .

This paper is devoted to the study of the initial value problem for some semilinear damped beam equation which has a non-negative definite energy functional. We prove a uniform bound and decay estimates of a solution for the equation with large initial data using the method introduced by Kawashima–Nakao–Ono [3].

We consider the Hardy–Henon equation − Δ u = | x | a u p with p > 1 and a ∈ R and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space R N . It has been conjectured that this property is true if (and only if) p p S ( a ) , where p S ( a ) is the Hardy–Sobolev exponent, given by ( N + 2 + 2 a ) / ( N − 2 ) . However, when N ⩾ 3 , the conjecture had up to now been proved only for a ⩽ 0 . Indeed the case a > 0 seems more difficult, due to p S ( a ) > ( N + 2 ) / ( N − 2 ) . In this paper, we prove the conjecture for a > 0 in dimension N = 3 , in the case of bounded solutions. Next, for the conjecture in the case a 0 , and for related estimates near isolated singularities and at infinity, we give new proofs – based in particular on doubling-rescaling arguments – and we provide some extensions of these estimates. These proofs are significantly simpler than the previously known ones. Finally, we clarify some of the previous results on a priori estimates for the related Dirichlet problem.

In this paper, we consider a class of second order hyperbolic systems of viscoelasticity in the whole space and show the global existence and sharp decay estimates of solutions under the smallness condition on the initial data. The tools that we use here are the time-weighted energy method and the semigroup argument.

We show that solutions of the initial boundary value problem for the coupled system of Kuramoto-Sivashinsky and Ginzburg-Landau (KS–GL) equations continuously depend on parameters of the system, and under some restrictions on parameters all solutions of initial boundary value problem for KS–GL equations tend to zero as (Formula presented.) with an exponential rate. © 2014 Taylor & Francis. 112T934 The work of V. K. Kalantarov was supported in part by The Scientific and Research Council of Turkey, [grant number 112T934].

118p; In this paper, we study the existence and regularity of the quasilinear parabolic equations: $$u_t-\text{div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu$$ in $\mathbb{R}^{N+1}$, $\mathbb{R}^N\times(0,\infty)$ and a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$. Here $N\geq 2$, the nonlinearity $A$ fulfills standard growth conditions and $B$ term is a continuous function and $\mu$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=\pm|u|^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with $B\equiv 0$, under minimal conditions on the boundary of domain and on nonlinearity $A$. Finally, due to these estimates, we solve the existence problems with $B(u,\nabla u)=|\nabla u|^q$ for $q>1$.

In this paper, we study the existence and regularity of the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu,$$ in either $\mathbb{R}^{N+1}$ or $\mathbb{R}^N\times(0,\infty)$ or on a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$ where $N\geq 2$. In this paper, we shall assume that the nonlinearity $A$ fulfills standard growth conditions, the function $B$ is a continuous and $\mu$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=\pm|u|^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with $B\equiv 0$, under minimal conditions on the boundary of domain and on nonlinearity $A$. Finally, due to these estimates, we solve the existence problems with $B(u,\nabla u)=|\nabla u|^q$ for $q>1$, Comment: 110 pages. Some typos corrected, to appear in the Memoirs of the AMS

We review the SBV regularity for solutions to hyperbolic systems of conservation laws and Hamilton-Jacobi equations. We give an overview of the techniques involved in the proof, and a collection of related problems concludes the paper.

The aim in this paper is to develop the time-weighted energy method for quasi-linear hyperbolic systems of viscoelasticity. As a consequence, we prove the global existence and decay estimate of solutions for the space dimension $n\geq 2$, provided that the initial data are small in the $L^{2}$-Sobolev space.

In the linear theory of thermoelasticity without energy dissipation, a dynamic Saint-Venant's principle is established for bounded and unbounded bodies. As a direct consequence, a uniqueness theorem is obtained which, for unbounded bodies, is free of any kind of a priori restriction concerning the growth of solutions at infinity.

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.

We consider radial solutions to the Cauchy problem for a linear wave equation with a small short-range electromagnetic potential (depending on space and time) and zero initial data. We present two dispersive estimates that provide, in particular, an optimal decay rate in time $t^{-1}$ for the solution. Also, we apply these estimates to obtain similar results for the linear massless Dirac equation perturbed by a potential.

In this article, the authors consider the linearized dynamic theory of a binary mixture consisting of an elastic solid and a gas. First, the continuous dependence of solutions on initial data and body forces is established. Then, a spatial decay estimate of an energetic measure associated to an admissible process is derived.