Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation
The global existence and global nonexistence of solutions of the Cauchy problem for wave equations with nonlinear damping \[ u_{tt}+ Q(t,x,u_t)-\Delta u= f(t,u),\quad (t,x)\in [0,\infty)\times \mathbb{R}^n \] are investigated. The authors establish various global nonexistence of weak solutions in terms of the function \(a(t,x)\) for which the inequality \(| Q(t,x,v)|\leq a(t,x)| v|^{m-1}\) holds, the right member \(f(x,u)\) and its integral for compactly supported initial data with negative or sufficiently negative initial energy, and the global continuability of local weak solutions in terms of the function \(\widetilde a(t,x)\) such that the inequality \(| Q(t,x,v)|\geq \widetilde a(t,x)| v|^{m-1}\) holds and the Fréchet derivative of \(f(t,u)\) with respect to \(u\). The results can be applied to the case \[ Q(x,t,u)= au_t| u_t|^{m-2},\quad f(x,u)=bu| u|^{p-2}. \]
negative or sufficiently negative initial energy
scientific article; zbMATH DE number 1246345