Critical exponent for damped wave equations with nonlinear memory

Ahmad Fino (1, 2)

1 LaMA--Liban - Laboratoire de mathématiques et applications
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the case when $1\leq n\leq3.$ Moreover, we derive a blow-up result under some positive data for in any dimensional space. It turns out that the critical exponent indeed coincides with the one to the corresponding semilinear heat equation.

Mots clés: Nonlinear damped wave equation; Global existence; Blow-up; Critical exponent; Large time asymptotic behavior

Domaines: Equations aux dérivées partielles [math.AP]

Ahmad Fino. Critical exponent for damped wave equations with nonlinear memory. 2010. ⟨hal-00473941v1⟩