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Zbl 1221.35242
Liu, Wenjun
Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source. (English)
[J] Topol. Methods Nonlinear Anal. 36, No. 1, 153-178 (2010). ISSN 1230-3429

The author investigates the initial-boundary value problem for nonlinear viscoelastic equations with strong damping and a nonlinear source $$\align u_{tt}-\Delta u +\int_0^t g(t-s)\Delta u(x,s)\,ds- \omega \Delta u_t+\mu u_t=|u|^{r-2}u, &\quad (x,t)\in \Omega\times (0,\infty),\\ u(x,0)=u_0(x),\ u_t(x,0)=u_1(x),\quad x\in \Omega, \qquad u(x,t)=0, &\quad (x,t)\in \partial\Omega\times [0,\infty),\endalign$$ where $\Omega$ is an open Lipschitz subset of $\Bbb R^n, n\ge 1,$ The relaxation function $g$ is a positive and decaying function. The initial data fulfil $u_0\in H_0^1(\Omega),\ u_1\in L^2(\Omega).$ Further assumptions are $\omega>0,\ \mu>\omega\lambda_1$, where $\lambda_1$ is the first eigenvalue of the operator $-\Delta$ under the homogeneous Dirichlet boundary condition, and $2<r<2^*=2n/(n-2)$ if $n\ge 3$, $2<r<\infty$ if $n=1,2$. The local existence of solutions is proved using the Faedo-Galerkin method and fixed point theorem. By virtue of the potential well theory and convexity technique, the global existence and polynomial decay to zero is verified if the initial data enter into the stable set. The solution blows up in a finite time if the initial data enter into the unstable set. Moreover the solution decays exponentially or polynomially to zero depending on the rate of the relaxation function.
[Igor Bock (Bratislava)]

MSC 2000:: *35L70 Second order nonlinear hyperbolic equations
35L15 Second order hyperbolic equations, initial value problems
35B35 Stability of solutions of PDE
35B40 Asymptotic behavior of solutions of PDE
74D05 Linear constitutive equations
35B44

Keywords: stable set; unstable set; potential well theory; convexity technique; polynomial decay

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