Blow up for nonlinear dissipative wave equations in \(\mathbb R^n\) (Q1766702)

This paper deals with the Cauchy problem for the following semilinear hyperbolic equation: \[ u_{tt}-\Delta u+|u_t|^{m-1}u_t=|u|^{p-1}u, \] with \(1\leq m<p\), \(p<\frac n{n-2}\), \(n\geq 3\). It is proved that for each \(\alpha>0\), \(\lambda\geq 0\) one can find infinitely many compactly supported initial data \(u_0, u_1\) having appropriately defined energy \(E=\lambda\) and \(\|\nabla u_0\|_2=\alpha\) but the solution \(u\) of the above mentioned Cauchy problem blows up in finite time.