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Global non-existence of solutions of a class of wave equations with non-linear damping and source terms. (English) Zbl 1073.35178

The authors prove nonexistence of global solutions in time of the initial boundary value problem

${u}_{tt}-{\Delta }{u}_{t}-{\text{div}\left(|\nabla u|}^{\alpha -2}\nabla u\right)-\text{div}\left(|\nabla {u}_{t}{|}^{\beta -2}\nabla {u}_{t}\right)+a|{u}_{t}{|}^{m-2}{u}_{t}=b{|u|}^{p-2}u,\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{4pt}{0ex}}t>0,$
$u\left(x,0\right)={u}_{0}\left(x\right),\phantom{\rule{4pt}{0ex}}{u}_{t}\left(x,0\right)={u}_{1}\left(x\right),\phantom{\rule{4pt}{0ex}}x\in {\Omega },\phantom{\rule{2.em}{0ex}}u\left(x,0\right)=0,\phantom{\rule{4pt}{0ex}}x\in \partial {\Omega },\phantom{\rule{4pt}{0ex}}t>0,$

where $a,b>0$, $\alpha ,\beta ,m,p>2$, and ${\Omega }$ is a bounded domain of ${ℝ}^{n}$ $\left(n\ge 1\right)$, with a smooth boundary $\partial {\Omega }$. This complicated equation is said to appear in nonlinear viscoelasticity theory. The authors suppose that there exist solutions in a function space $Z$, where

$Z={L}^{\infty }\left(\left[0,T\right);{W}_{0}^{1,\alpha }\left({\Omega }\right)\right)\cap {W}^{1,\infty }\left(\left[0,T\right);{L}^{2}\left({\Omega }\right)\right)\cap {W}^{1,\beta }\left(\left[0,T\right);{W}_{0}^{1,\beta }\left({\Omega }\right)\right)\cap {W}^{1,m}\left(\left[0,T\right);{L}^{m}\left({\Omega }\right)\right)$

for $T>0$. Local existence and uniqueness are not especially discussed. The energy is defined by

$E\left(t\right)=\frac{1}{2}{\int }_{{\Omega }}{u}_{t}^{2}\phantom{\rule{0.166667em}{0ex}}dx+\frac{1}{\alpha }{\int }_{{\Omega }}{|\nabla u|}^{\alpha }\phantom{\rule{0.166667em}{0ex}}dx-\frac{b}{p}{\int }_{{\Omega }}{|u|}^{p}\phantom{\rule{0.166667em}{0ex}}dx·$

Let ${r}_{\alpha }$ denote the Sobolev critical exponent of ${W}_{0}^{1,\alpha }\left({\Omega }\right)$. There is just one theorem.

Theorem: Assume that $\beta <\alpha$ and $max\left\{m,\alpha \right\}. If $E\left(0\right)<0$, then the solution $u\in Z$ cannot exist for all time.

The authors insist that their result improves one by Z. Yang [Math. Methods Appl. Sci. 25, No. 10, 825–833 (2002; Zbl 1009.35051)], whose result requires an additional assumption, $H\left(0\right)>A$ (a constant depending on the size of ${\Omega }$).

The proof is performed as follows. Define

$L\left(t\right)={H}^{1-\sigma }\left(t\right)+\epsilon {\int }_{{\Omega }}u{u}_{t}\phantom{\rule{0.166667em}{0ex}}dx,$

where $H\left(t\right)=-E\left(t\right)$. The main aim of the proof is to show that for suitable $\epsilon$, $\sigma >0$

${L}^{\text{'}}\left(t\right)\ge \xi {L}^{q}\left(t\right),\phantom{\rule{1.em}{0ex}}q>1·$

It follows easily from this inequality that $L\left(t\right)$ attains $\infty$ in finite time. The mathematical tool of higher grade are not needed, but silful calculation is performed to obtain the inequality. Young’s inequality, and Sobolev’s inequality, Poincarés inequaly in bounded domains are exploited.

Global existence is not especially referred.

##### MSC:
 35L75 Nonlinear hyperbolic PDE of higher $\left(>2\right)$ order 35L35 Higher order hyperbolic equations, boundary value problems