A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale‐invariant lower order terms

Abstract

In this note, two blow‐up results are proved for a weakly coupled system of semilinear wave equations with distinct scale‐invariant lower order terms both in the subcritical case and in the critical case when the damping and the mass terms make both equations in some sense “wave‐like.” In the proof of the subcritical case, an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case, we employ a test function‐type method that has been developed recently by Ikeda‐Sobajima‐Wakasa and relies strongly on a family of certain self‐similar solutions of the adjoint linear equation. Therefore, as critical curve in the p−q plane of the exponents of the power nonlinearities for this weakly coupled system, we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.