Wave equations with super-critical interior and boundary nonlinearities (Q435810)

In this article is considered the semi-linear boundary value problem NEWLINE\[NEWLINE \begin{aligned} &u_{tt}-\Delta u +g_0(u_t)=f(u)\quad {\text{ in}}\quad \Omega\times [0, \infty),\\ &\partial_{\nu}u+u+g(u_t)=h(u)\quad {\text{ on}}\quad \Gamma\times [0, \infty),\\ &u(0, x)=u_0(x)\in H^1(\Omega), u_t(0, x)=u_1(x)\in L^2(\Omega), \end{aligned} \tag{1} NEWLINE\]NEWLINE where \(\Omega\subset\mathbb R^n\) is a bounded open set with sufficiently smooth boundary \(\Gamma\). Here \(g_0\) and \(g\) model respectively interior and boundary damping terms, \(f\) and \(h\) represent source terms.NEWLINENEWLINEThe authors give an overview of results for \((1)\) which are connected with Hadamard local well possedness, global existence, blow-up and non-existence theorems, also estimates of the uniform energy dissipation rates for appropriate classes of solutions.