Critical Exponents Curve for Semilinear System of Weakly Coupled Effectively Damped Waves with Different Power nonlinearities

A. Mohammed Djaouti

Kragujevac Journal of Mathematics, Tome 49 (2025) no. 3, p. 375

Résumé

In this paper we prove a blow-up result for the semi linear system of weakly coupled effectively damped waves with different power nonlinearities \begin{align*} \begin{split} {tt}-\Delta u+b(t)u_{t}=|v|^{p}, \quad v_{tt}-\Delta v+b(t)v_{t}= |u|^{q}, (0,x)=u_{0}(x),\quad u_{t}(0,x)=u_{1}(x),\quad v(0,x)=v_{0}(x),\quad v_{t}(0,x)=v_{1}(x), \end{split} \end{align*} where $b(t)$ will be explained in detail in the next sections. We apply the so called ``test function method'' to determine the range for the exponents $p,q>0$ in the nonlinear terms in which local in time existence may not globally prolonged with respect to the $t$ variable under suitable integral sign assumptions for the Cauchy data $u_0,u_1,v_0,v_1$. Since we prove the blow-up in a complementary range for powers of the nonlinear terms to that for the global existence of small data solutions (see \cite{DjaoutiReissig}), the main blow-up of this paper is optimal.

Détail

Classification : 35L52, 35L71
Keywords: Weakly coupled hyperbolic systems, damped wave equations, Cauchy problem, blow up, effective dissipation