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.

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• Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities
• A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case
• The semilinear Euler-Poisson-Darboux equation: a case of wave with critical dissipation
• A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type
• Blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime
• Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part
• Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms
• Blow-up of solutions to Nakao's problem via an iteration argument
• Integral representation formulae for the solution of a wave equation with time-dependent damping and mass in the scale-invariant case
• New blow-up result for the weakly coupled wave equations with a scale-invariant damping and time derivative nonlinearity