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.


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