Global existence and asymptotic stability for viscoelastic problems.(English)Zbl 1015.35071

The authors investigate existence, uniqueness and decay rates of a solution of the semilinear viscoelastic wave equation $u_{tt}-\Delta u+\alpha u + f(u) + \int_0^t g(t-\tau)\Delta u(\tau) d\tau + h(u_t)=0 \quad \text{in} \Omega\times (0,\infty),$ $$u(x,0)=u_0(x)$$, $$u_t(x,0)=u_1(x)$$, $$x\in \Omega$$, where $$\Omega$$ is any bounded or finite measure domain of $$\mathbb R^n$$. If $$\Omega$$ has the nonempty boundary $$\Gamma$$, then it is assumed to be regular only in the case of the boundary condition $$u=0$$ on $$\Gamma$$. The functions $$f, h$$ are of the form $$f(s)=\gamma|s|^\xi s$$, $$h(s)=\beta |s|^\rho s$$, where $$0<\xi$$, $$\rho\leq \frac 2{n-2}$$ if $$n\geq 3$$ and $$\Omega$$ is bounded or $$\Omega$$ is unbounded but possesses finite measure, $$0<\xi=\rho= \frac 2{n-2}$$ if $$n\geq 3$$ and $$\Omega$$ is unbounded, $$\xi,\rho>0$$ if $$n=1,2$$ and $$\Omega$$ is arbitrary. The kernel function $$g\in C^2(\mathbb R_+)$$ is bounded and satisfying $$1-\int_0^\infty g(s)ds= l>0$$, $$-\xi_1g(t)\leq g'(t)\leq -\xi_2 g(t)$$, $$0\leq g''(t)\leq \xi_3 g(t)$$ $$\forall t\geq 0.$$ Let $$\mathcal H$$ is a Hilbert space endowed with the inner product $$(u,v)_\mathcal H = (u,v)_{H^1_0(\Omega)}+(\Delta u,\Delta v).$$
The main results are the following theorems:
Theorem 2.1. Let $$\{u_0,u_1\}\in \mathcal H\times H^1_0(\Omega)$$. Then the above problem possesses a unique regular solution in the class $$u,u'\in L^\infty(0,\infty;H^1_0(\Omega))$$, $$u''\in L^\infty(0,\infty;L^2(\Omega))$$. Moreover, if $$\|g\|_{L^1(0,\infty)}$$ is sufficiently small and $$\Omega$$ has finite measure, the energy $$E(t)=\frac 12\int_\Omega\{|u_t(x,t)|^2+|\nabla u(x,t)|^2+\alpha |u(x,t)|^2+\frac \gamma{\xi+2}|u(x,t)|^2\} dx$$ has the decay rates $$E(t)\leq (\varepsilon\theta t+[E(0)]^{-\rho/2})^{-2/\rho}$$, $$\forall t\geq 0$$, $$\forall \varepsilon\in (0,\varepsilon_0]$$, $$\theta>0$$, $$\varepsilon_0>0.$$
Theorem 2.2. Let $$\{u_0,u_1\}\in H^1_0(\Omega)\times L^2(\Omega)$$. Then the above problem possesses a unique weak solution in the class $$u\in C^0([0,\infty);H^1_0(\Omega))\cap C^1([0,\infty);L^2(\Omega))$$ with the same decay as in the Theorem 2.1.

MSC:

 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 74D10 Nonlinear constitutive equations for materials with memory 93D20 Asymptotic stability in control theory