Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms

Abstract

In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the p-q plane for the pair of exponents (p,q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we provide a different approach from the test function method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of the solution on the critical curve, improving in some cases the upper bound estimate for the lifespan. More precisely, we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

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Cited by in corpus (5)

• A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms
• 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
• Lifespan estimates for local solutions to the semilinear wave equation in Einstein-de Sitter spacetime
• Blow-up and lifespan estimates for Nakao's type problem with nonlinearities of derivative type