Lifespan of solutions to the damped wave equation with a critical nonlinearity
The main result of this paper is an inequality of the form \(T(\varepsilon)\leq\exp(C\varepsilon^{-p})\) for the blow-up time for a mild solution of NEWLINE\[NEWLINE u_{tt}-\Delta u + u_{t}=|u|^p, NEWLINE\]NEWLINE where the spatial domain is all of \(\mathbb R^n\), with \(n\geq 1\), \(p=1+2/n\), and the data have the form \((\varepsilon u_0, \varepsilon u_1)\), with \((u_0,u_1)\in H^1\times L^2\), \(u_0+u_1\in L^1\), and \(\int_{\mathbb R^n}(u_0+u_1)dx>0\). The method is similar to that of \textit{M. Kirane} and \textit{M. Qafsaoui} [Adv. Nonlinear Stud. 2, No. 1, 41--49 (2002; Zbl 1056.35020)] and \textit{Q. S. Zhang} [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 2, 109--114 (2001; Zbl 1056.35123)].
scientific article; zbMATH DE number 6582628