Critical exponent for a nonlinear wave equation with damping
The critical exponent for the equation \(u_{tt}+u_t=\Delta u +|u|^p\) in \({\mathbb R}^n\) is found. More precisely, it is shown that if \(p>1+2/n\) then all small data solutions are global while all solutions with \(\int u(0,x) dx>0\), \(\int u_t(0,x) dx>0\) blow up in finite time when \(1<p<1+2/n\). This means that the critical exponent is the same as for the equation \(u_t=\Delta u +|u|^p\). The decay rate of small data global solutions is also studied for \(p>1+2/n\).
scientific article; zbMATH DE number 1656000