Systems of semilinear wave equations with multiple speeds in two space dimensions and a weaker null condition

Abstract

<p style='text-indent:20px;'>We consider the Cauchy problem for systems of semilinear wave equations with multiple propagation speeds in two space dimensions. The nonlinear terms are assumed to be cubic and to depend only on first derivatives of the unknowns. The null condition is one of the sufficient condition for the small data global existence for the single-speed case, and it is extended to the multiple-speed case by many authors. Recently, various weaker sufficient conditions for the small data global existence were obtained for the single-speed case. In this paper, we introduce the multiple-speed version of one of such weaker null conditions, and we show the small data global existence. The asymptotic behavior of global solutions is also investigated through associated systems of ODEs called the reduced systems.</p>