Stabilization of the semilinear wave equation (Q2774041)

Two local stabilisation theorems for the nonlinear wave system NEWLINE\[NEWLINE\square u+a(x) \partial_tu+f(u)= 0\text{ on }]0,+\infty [\times\OmegaNEWLINE\]NEWLINE \(( \Omega \subseteq \mathbb{R}^d)\) are proved in the form \(E(u)(t)\leq ce^{-\gamma t} E(u)(0)\), \(t\geq 0\), where \(E(u)(t)\) is the usual energy for the linearised system perturbed by the term \(\int_\Omega F(u(t,x))dx\) where \(F(u)=\int^u_0 f(s)ds\). The nonlinearity is superlinear at infinity and is assumed to satisfy \(sf(s)\geq 0\), \(s\in\mathbb{R}\), \(|f^{(j)}(s) |\leq c(1+|s|)^{p-j}\), \(j=0,1\), \(c>0\), \(p\geq 1\), \((d-2)p\leq d\). The first result requires a geometric condition on the boundary of \(\Omega\) and the second relaxes this condition in the subcritical case \((1\leq p<d/(d-2))\) to a geometric control condition and a unique prolongation assumption on a related linear system. The results require the use of microlocal techniques of partial differential equations, in the form of Radon measures.