Variational inequality for quasilinear wave equations with nonlinear damping terms
The paper deals with the unilateral problem \(u_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta u+\delta |u_t|^p u_t\geq \mu |u|^\alpha u ,\) \(v_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta v+\delta |v_t|^p v_t\geq \mu |v|^\alpha v\), \(u\geq 0\), \(v\geq 0,\) \((u_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta u+\delta |u_t|^p u_t- \mu |u|^\alpha u)u = 0 ,\) \((v_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta v+\delta |v_t|^p v_t- \mu |v|^\alpha v)v=0\), \(\text{in} Q,\) \(u=v=0 \;\text{in} \Sigma\), \(u(0)=u_0\geq 0\), \(v(0)=v_0\geq 0\), \(u_t(0)-u_1\geq 0\), \(v_t(0)-v_1\geq 0\), \((u_t(0)-u_1)u_0=0\), \((v_t(0)-v_1)v_0=0 \;\text{in} \Omega,\) where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\Gamma\), \(T>0\), \(Q=\Omega \times (0,T), \Sigma=\Gamma\times (0,T)\), \(M(s)=a+bs\), \(a+b\geq 0, b\geq 0\), \(\alpha>0\), \(\rho\geq 0\), \(\delta>0\), \(\mu\in \mathbb R.\) The authors formulate the penalized problem and employ the Galerkin method to construct its solution. They derive a priori estimates and obtain the existence and uniqueness of a penalized solution. The existence of a weak solution \(\{u,v\}\in L^\infty(0,T;H^1_0(\Omega)\cap H^2(\Omega))^2\), \(\{u',v'\}\in [L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H_0^1(\Omega))]^2\;\) of the original unilateral problem is obtained by performing the limit of the subsequence of the penalized solutions \(\{u_\varepsilon,v_\varepsilon\}\).
scientific article; zbMATH DE number 1788057