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Article Dans Une Revue Journal of Mathematical Analysis and Applications Année : 2020

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

Résumé

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.

Dates et versions

hal-02356801 , version 1 (09-11-2019)

Identifiants

Citer

Mohamed Ali Hamza, Hatem Zaag. The blow-up rate for a non-scaling invariant semilinear wave equations. Journal of Mathematical Analysis and Applications, 2020, 483 (2), pp.123652. ⟨10.1016/j.jmaa.2019.123652⟩. ⟨hal-02356801⟩
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