Blow-up of solutions to Nakao's problem via an iteration argument

Abstract

In this paper, we consider blow-up behavior of weak solutions to a weakly coupled system for a semilinear damped wave equation and a semilinear wave equation in ${\mathbb{R}}^n$. This problem is part of the so-called Nakao's problem proposed by Professor Mitsuhiro Nakao (Kyushu University) for a critical relation between the exponents $p$ and $q$. By applying an iteration method for unbounded multipliers with a slicing procedure, we prove blow-up of weak solutions for Nakao's problem even for small data. We improve the blow-up result and upper bound estimates for lifespan comparing with the previous research, especially, in higher dimensional cases.

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Cited by in corpus (4)

• On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups
• A note on blow-up results for semilinear wave equations in de Sitter and anti-de Sitter spacetimes
• A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type
• Blow-up and lifespan estimates for Nakao's type problem with nonlinearities of derivative type