Lifespan of solutions to the damped wave equation with a critical nonlinearity. (English) Zbl 1361.35114
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
\[
u_{tt}-\Delta u + u_{t}=|u|^p,
\]
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 M. Kirane and M. Qafsaoui [Adv. Nonlinear Stud. 2, No. 1, 41–49 (2002; Zbl 1056.35020)] and Q. S. Zhang [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 2, 109–114 (2001; Zbl 1056.35123)].
Reviewer: Satyanad Kichenassamy (Reims)
MSC:
| 35L71 | Second-order semilinear hyperbolic equations |
| 35B44 | Blow-up in context of PDEs |
| 35L15 | Initial value problems for second-order hyperbolic equations |
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