Existence of solutions for a variational unilateral system (Q2778489)

Let \(\Omega\) be a bounded and open set of \(\mathbb{R}^n\) with smooth boundary \(\Gamma=\partial\Omega\), and let \(T\) be a positive real number. Let \(Q= \Omega\times (0,T)\) be the cylinder with lateral boundary \(\Sigma=\Gamma \times (0,T)\). The authors study the existence of weak solutions of the nonlinear unilateral mixed problem associated to the inequalities NEWLINE\[NEWLINE\begin{gathered} u_{tt}-M\bigl( |\nabla u|^2 \bigr)\Delta u+\theta \geq f\quad\text{in }Q,\\ \theta_t-\Delta \theta+ u_t\geq g\quad\text{in }Q,\\ u=\theta=0\quad\text{in }\Sigma,\\ u(t)-u_0,\;u'(0) =u_1,\;\theta (0)=\theta_0, \end{gathered}NEWLINE\]NEWLINE under appropriate assumptions on \(M\), \(f\) and \(g\). They employ Galerkin's approximation method and the penalization method.