# The blow-up rate for a non-scaling invariant semilinear wave equations

Abstract : We consider the semilinear wave equation $\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb{R}$. We show an upper bound for any blow-up solution of (1). Then, in the one space dimensional case, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, namely $u'' =|u|^{p-1}u\log^a (2+u^2)$ Unlike the pure power case ($g(u)=|u|^{p-1}u$) the difficulties here are due to the fact that equation (1) is not scale invariant.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-02356801
Contributor : Hatem Zaag <>
Submitted on : Saturday, November 9, 2019 - 6:10:43 AM
Last modification on : Tuesday, November 19, 2019 - 1:45:18 AM

### Identifiers

• HAL Id : hal-02356801, version 1
• ARXIV : 1906.12059

### Citation

Mohamed Ali Hamza, Hatem Zaag. The blow-up rate for a non-scaling invariant semilinear wave equations. Journal of Mathematical Analysis and Applications, Elsevier, In press. ⟨hal-02356801⟩

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