Lyapunov functional and blow-up results for a class of perturbations of semilinear wave equations in the critical case (Q2911982)

The blow-up solutions for the following semilinear wave equation are investigated: NEWLINE\[NEWLINE \begin{aligned} & u_{tt}=\Delta u +|u|^{p-1}u+f(u)+g(\partial_t u),\quad (x, t)\in \mathbb R^n\times [0, T),\\ & (u(x, 0), u_t(x, 0))=(u_0(x), u_1(x))\in H^1_{{\text{ loc}}, u}(\mathbb R^n)\times L^2_{{\text{ loc}}, u}(\mathbb R^n), \end{aligned}NEWLINE\]NEWLINE where \(p=1+{4\over {n-1}}\), \(n\geq 2\), \(f\) and \(g\) are locally Lipschitz-continuous satisfying the conditions \(|f(u)|\leq M\Bigl(1+|u|^q\Bigr)\), \(q<p\), \(M>0\), \(|g(u)|\leq M(1+|u|)\). The authors introduce a Lyapunov functional for this problem and prove that the blow-up rate of any singular solution is determined of the solution of the nonperturbed associated ODE.