On the Strongly Damped Wave Equation: $u_{tt} - \Delta u - \Delta u_t + f(u) = 0$

Abstract

The following initial boundary value problem is considered \[ \begin{gathered} u_{tt} - \Delta u - \lambda \Delta u_t + f(u) = 0,\quad (x,t) \in \Omega \times ] {0,T} [,\quad \lambda > 0, \hfill \\ u = 0\quad {\text{on }}\partial \Omega \times [0,T), \hfill \\ u(x,0) = w_0 (x),\qquad u_t (x,0) = w_1 (x) \hfill \\ \end{gathered} \] where $\Omega $ is a bounded domain in $R^N $ with a sufficiently regular boundary $\partial \Omega $. In Part 1, a theorem on local existence and uniqueness is proved for $w_0 $ in $H_0^1 (\Omega )$ and $w_1 $ in $L^2 (\Omega )$, under a certain Lipschitzian condition on f. In Part 2, the question of global existence and asymptotic behavior for $t \to \infty $ is studied, under more restrictive conditions, namely $1 \leqq N \leqq 3$, $f \in C^1 (\mathbb{R},\mathbb{R})$, $f(0) = 0$, and $f' \geqq - c$ with $c > 0$ “small” and $w_0 \in H_0^1 (\Omega ) \cap H^2 (\Omega )$, $w_1 \in L^2 (\Omega )$. It is proved that under these conditions, a unique solution $u(t)$ exists on $\mathbb{R}_ + $ such that $\| {u_t (t)} \|$ and $\| {\Delta u (t)} \|$ decay exponentially to 0 as $t \to \infty $. ($\| \cdot \|$ denotes the $L^2 (\Omega )$ norm.) The method followed in this paper is that of successive linearizations (Part 1) and Galerkin (Part 2).