Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials

In this paper, we observe how the heat equation in a noncylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case [Alvarez-Caudevilla and Lemenant, Adv. Differ. Equ. 15 (2010) 649-688]. We provide a strong convergence result for the solution by use of energetic methods and Γ-convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument due to B. Simon.

On some boundary value problems for the heat equation in a non-regular type of a prism of ℝN+1

This paper is devoted to the analysis of the boundary value problem {\partial_{t}u-\Delta u=f}, with an N-dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain. The problem is settled in a noncylindrical domain of the form Q=\{(t,x_{1})\in\mathbb{R}^{2}:0<t<T,\varphi_{1}(t)<x_{1}<\varphi_{2}(t)\}% \times\prod_{i=1}^{N-1}{]0,b_{i}[}, where {\varphi_{1}} and {\varphi_{2}} are smooth functions. One of the main issues of the paper is that the domain can possibly be non-regular; for instance, the significant case when {\varphi_{1}(0)=\varphi_{2}(0)} is allowed. We prove well-posedness results for the problem in a number of different settings and under natural assumptions on the coefficients and on the geometrical properties of the domain. This work is an extension of the one-dimensional case studied in [4].

Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>In this paper we investigate the existence and uniqueness of solution for a initial boundary value problem for the following nonlinear wave equation: </span></p><p><span>u′′</span> − ∆ u + | u | ˆρ = f in Q</p><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>where </span><span>Q </span><span>represents a non-cylindrical domain of </span><span>R^{</span><span>n</span><span>+</span><span>1}</span><span>. The methodology, cf. Lions [3], consists of transforming this problem, by means of a perturbation depending on a parameter </span><span>ε &gt; </span><span>0, into another one defined in a cylindrical domain </span><span>Q </span><span>containing </span><span>Q</span><span>. By solving the cylindrical problem, we obtain estimates that depend on </span><span>ε</span><span>. These ones will enable a passage to the limit, when </span><span>ε </span><span>goes to zero, that will guarantee, later, a solution for the non-cylindrical problem. The nonlinearity </span><span>|</span><span>u_</span><span>ε</span><span>|^</span><span>ρ </span><span>introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar [8] plus a contradiction process. </span></p></div></div></div></div></div></div>

Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

We study Kirchhoff plates equations with viscoelastic boundary conditions in a noncylindrical domain. This work is devoted to proving the global existence, uniqueness of solutions, and decay of the energy of solutions for Kirchhoff plates equations in a non-cylindrical domain.

TWO-DIMENSIONAL INCOMPRESSIBLE IDEAL FLOWS IN A NONCYLINDRICAL MATERIAL DOMAIN

The purpose of this work is to prove the existence of a weak solution of the two-dimensional incompressible Euler equations on a noncylindrical domain consisting of a smooth, bounded, connected and simply connected domain undergoing a prescribed motion. We prove the existence of a weak solution for initial vorticity in Lp, for p > 1. This work complements a similar result by C. He and L. Hsiao, who proved existence assuming that the flow velocity is tangent to the moving boundary, see Ref. 6.