A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type

Abstract

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied.

References in corpus (18)

• From $p_0(n)$ to $p_0(n+2)$
• The Threshold between Effective and Noneffective Damping for Semilinear Waves
• Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent
• Test function method for blow-up phenomena of semilinear wave equations and their weakly coupled systems
• Blow-up for semilinear damped wave equations with sub-Strauss exponent in the scattering case
• Fujita versus Strauss - a never ending story
• The Glassey conjecture with radially symmetric data
• Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity
• Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity
• A global existence result for a semilinear wave equation with scale-invariant damping and mass in even space dimension
• Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities
• Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces
• A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms
• Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case
• Almost global solutions of semilinear wave equations with the critical exponent in high dimensions
• 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
• The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

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• Blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime
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• A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type
• Integral representation formulae for the solution of a wave equation with time-dependent damping and mass in the scale-invariant case
• A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative-type
• Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type
• Blow-up and lifespan estimates for Nakao's type problem with nonlinearities of derivative type