Exponential stability for the wave model with localized memory in a past history framework
The article investigates the well-posedness as well as the exponential decay of solutions to a mixed problem of a locally distributed viscoelastic damped wave equation in the variable u(x,t), with past history. Introducing a new variable \(\eta^t(x,s) = u(x,t)-u(x,t-s)\) the model is transformed into a mixed problem for a system in variables \(u\) and \(\eta\), \[ \begin{aligned} &{\rho}(x) u_{tt}-\operatorname{div}[k(x)\nabla u]-\int_0^{\infty}g(s)\operatorname{div}a(x)\nabla{\eta}^t(.,s)]\,\mathrm ds+b(x)u_t=0,\quad \text{in }\Omega \times (0,\infty)\\ &\eta_t=-\eta_s+u_t \qquad\text{in }\Omega \times(0,\infty)\times (0,\infty),\\ &u=0 \quad \text{ on } \partial\Omega \times (0,\infty), \;\eta=0 \quad \text{ on } \partial\Omega \times (0,\infty),\\ &u(x,0)=u_0(x), \; u_t(x,0)=u_1(x), \\ &\eta^0(x,s)=\eta_0(x,s), \qquad x\in\Omega, \;s\in[0,\infty), \\ &\eta^t(x,0) =0, \qquad x\in\Omega, \;t\in[0,\infty) \end{aligned} \] To this system the energy functional \({E_{u,{\eta}}}(t)\) is associated. In Section 2, after a quite extensive review of the literature on well-posedness and decay properties of solutions to the wave equation, under some assumptions, a well-posedness result for the above system is proved. From it, a well posedness result for the initial mixed problem is deduced. In the third section, under additional assumptions, an exponential stability result is proved and one investigates particular cases when system is exponentially stable in the absence of the frictional damping term. The forth section is devoted to a result related to the local uniform decay associated with the semi-linear model, which is the above model, where a nonlinear source \(f(u)\) has been introduced. The fifth section is the Appendix and contains some additional definitions and results that might be used in the proofs of exponential stability.
locally distributed viscoelastic damping
scientific article; zbMATH DE number 6852245
scientific article