Exponential decay for the quintic wave equation with locally distributed damping (Q6634485)

It is considered the quintic wave equation with locally distributed damping, \(u_{tt}-\Delta u+u^5+a(x)u_t=0\), on a smooth bounded domain \(\Omega\subset\mathbb{R}^3\), with homogeneous Dirichlet boundary condition and nonnegative \(a\in L^\infty\). The authors study the well-posedness and the (exponential) stabilization of solutions. For initial data \((u(0),\partial_t u(0))\in H^1_0\times L^2\) it is proved that there exists a unique weak solution in \(C_t H^1_0\times C_t^1 L^2\times L^4_{t, \mathrm{local}}L^{12}_x\). Under the condition that \(a\in C^1(\bar\Omega)\), \(a\) is nontrivial near the boundary, and that geometric control condition is satisfied, an observability inequality is obtained and so is the exponential decay of the energy. To prove observability inequalities, the authors approximate weak solutions by regular solutions and prove the essential observability inequality. The treatment of these approximate solutions requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures.