A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations (Q2918667)

The initial value problem for the equation \(u_{tt}=\Delta u+| u| ^{p-1}u+f(u)+g(u_t),\) where \(t>0\), \(x\in \mathbb{R}^N\), \(| f(u)| \leq C(1+| u| ^q)\), \(| g(v)| \leq C(1+| v| )\), \(q<p<p_c\equiv 1+4/(N-1)\) is dealt with. The authors derive a Lyapunov functional in similarity variables, which is crucial step for deriving the blow-up rate. To prove the blow-up rate of the solution they use the interpolation in the Sobolev spaces, some Gagliardo-Nierenberg estimates and a covering technique adapted to the geometric shape of the blow-up surface.