On the global solvability of solutions to a quasilinear wave equation with localized damping and source terms

Abstract

We prove existence and uniform stability of strong solutions to a quasilinear wave equation with a locally distributed nonlinear dissipation with source term of power nonlinearity of the type $u^{″}-M\left({\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\right)\Delta u+a\left(x\right)g\left(u^{\prime }\right)+f\left(u\right)=0,$ in $\Omega \times \left]0,+\infty \right[$, $u=0,$ on $\Gamma \times \left]0,+\infty \right[,$ $u\left(x,0\right)=u_0\left(x\right),u^{\prime }\left(x,0\right)=u_1\left(x\right)$, in $\Omega$.