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On the strongly damped wave equation. (English) Zbl 1068.35077

The author considers the strongly damped wave equation

$\begin{array}{c}{u}_{tt}-\omega {\Delta }{u}_{t}-{\Delta }u+\phi \left(u\right)=f\left(t\right),\phantom{\rule{4pt}{0ex}}x\in {\Omega },\phantom{\rule{4pt}{0ex}}t>0,\phantom{\rule{4pt}{0ex}}u\left(x,0\right)={u}_{0}\left(x\right),\\ {u}_{t}\left(x,0\right)={u}_{1}\left(x\right),\phantom{\rule{4pt}{0ex}}u\left(x,t\right)=0\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}x\in \partial {\Omega }·\end{array}\phantom{\rule{2.em}{0ex}}\left(2\right)$

Here ${\Omega }\subseteq {ℝ}^{3}$ is a smooth, bounded domain, and $\omega >0$. The emphasis of the paper is on the growth order of the nonlinearity which may be 5. A functional setting is imposed on (1), i.e. one sets $A=-{\Delta }$ on $\text{dom}\left(A\right)={H}^{2}\left({\Omega }\right)\cap {H}_{0}^{1}\left({\Omega }\right)$ and then one defines the phase space of (1) in terms of the fractional power spaces $D\left({A}^{s}\right)$, i.e. one sets ${H}_{s}=D\left({A}^{\left(1+s\right)/2}\right)×D\left({A}^{s/2}\right)$. The assumptions on the nonlinearity are as follows:

$|\phi \left(s\right)-\phi \left(r\right)|\le c|r-s|\left(1+|r{|}^{4}+|s{|}^{4}\right)·\phantom{\rule{2.em}{0ex}}\left(1\right)$

One assumes that there is a decomposition $\phi ={\phi }_{0}+{\phi }_{1}$, ${\phi }_{j}\in C\left(ℝ\right)$ such that

$|{\phi }_{0}{\left(r\right)|\le c\left(1+|r|\right)}^{5},\phantom{\rule{4pt}{0ex}}{\phi }_{0}\left(r\right)r\ge 0,\phantom{\rule{4pt}{0ex}}|{\phi }_{1}{\left(r\right)|\le c\left(1+|r|}^{\gamma }\right),\phantom{\rule{2.em}{0ex}}\left(3\right)$

some $\gamma \in \left(0,5\right)$ and one also assumes $lim inf{r}^{-1}{\phi }_{1}\left(r\right)>{\alpha }_{1}$ as $|r|\to \infty$, where ${\alpha }_{1}>0$. Based on these assumptions, Theorem 1 asserts the existence of a unique global solution to (1), what entails the existence of a strongly continuous solution semigroup $S\left(t\right)$, $t>0$, which has suitable Lipschitz properties by Theorem 2. Theorem 3 then asserts the existence of an absorbing set. Based on this fact, the existence of a universal attractor follows (Theorem 4). The author then considers a subcritical case which arises if $f\in {L}^{2}\left({\Omega }\right)$ is $t$-independent and where the ${\phi }_{j}$ in (2) satisfy:

${\phi }_{0}=0,\phantom{\rule{4pt}{0ex}}{\phi }_{1}\in {C}^{1}\left(ℝ\right),\phantom{\rule{4pt}{0ex}}|{\phi }_{1}^{\text{'}}{\left(r\right)|\le c\left(1+|r|}^{\gamma -1}\right),\phantom{\rule{4pt}{0ex}}r\in ℝ·\phantom{\rule{2.em}{0ex}}\left(4\right)$

Under assumptions (4), further properties of the global attractor can be deduced. In addition, the existence of exponential attractors can be proved, as is shown in the last part of the paper.

##### MSC:
 35L75 Nonlinear hyperbolic PDE of higher $\left(>2\right)$ order 37L30 Attractors and their dimensions, Lyapunov exponents 35B41 Attractors (PDE)
##### References:
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