General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history
The author investigates the initial-boundary value problem for a viscoelastic wave equation with dynamic boundary conditions, source term and a nonlinear weak damping realized on a part of the boundary and a past history: \[ \begin{aligned} &u_{tt}(t)-\Delta(u+u_t)(t)+\int_0^\infty g(s)\Delta u(t-s)\,\text{d}s=0\;\text{ in }\Omega\times \mathbb{R}^+, \;u(t)=0\;\text{ on }\Gamma_0\times \mathbb{R}^+,\\ &u_{tt}(t)= - \frac{\partial (u+u_t)}{\partial \nu}+\int_0^\infty g(s)\frac{\partial u}{\partial \nu}(t-s)\,\text{d}s-h(u_t)-f(u)\;\text{ on } \Gamma_1\times \mathbb{R}^+, \\ &u(x,-t)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \Omega, \end{aligned} \] where \(\Omega\subset \mathbb{R}^n (n\geq 1)\) is a regular and bounded domain, \(\partial \Omega =\Gamma_0\cup \Gamma_1,\;\Gamma_0\cap \Gamma_1=\emptyset.\) Using a new variable \(\eta(x,t,s)=u(t)-u(t-s)\) and assuming \(1-\int _0^\infty = \ell>0\) the problem is transformed to \[ \begin{aligned} &u_{tt}(t)-\ell\Delta u(t)-\Delta u_t(t)-\int_0^\infty g(s)\Delta \eta(s)\,\text{d}s=0\;\text{ in }\Omega\times \mathbb{R}^+, \;u(t)=0\;\text{ on } \Gamma_0\times \mathbb{R}^+,\\ &u_{tt}(t)= - \ell\frac{\partial u}{\partial \nu}-\frac{\partial u_t}{\partial \nu}-\int_0^\infty g(s)\frac{\partial \eta}{\partial \nu}(s)\,\text{d}s-h(u_t)-f(u)\;\text{ on } \Gamma_1\times \mathbb{R}^+, \\ &u(x,-t)=u_0(x),\;u_t(x,0)=u_1(x),\;\eta_0=u_0(x)-u_0(x,-s),\;x\in \Omega. \end{aligned} \] The main result is the general decay of energy to the transformed problem.
scientific article; zbMATH DE number 6909942
scientific article