Test function method for blow-up phenomena of semilinear wave equations and their weakly coupled systems

Abstract

In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function method and give a simple proof of the derivation of sharp upper bound of lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical case, we use a family of self-similar solution to the standard wave equation including Gauss's hypergeometric functions which are originally introduced by Zhou (1992). However, our framework is much simpler than that. As a consequence, we found new $(p,q)$-curve for the system ${\partial }_t^{2}u-\Delta u=|v{|}^q$, ${\partial }_t^{2}v-\Delta v=|{\partial }_{t}u{|}^p$ and lifespan estimate for small solutions for new region.

31 pages

References in corpus (4)

• The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions
• Upper bound for lifespan of solutions to certain semilinear parabolic, dispersive and hyperbolic equations via a unified test function method
• The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions
• Life span of small solutions to a system of wave equations

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• Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case
• Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity
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• Threshold for the existence of scattering states for nonlinear Schrödinger equations without gauge invariance
• Slicing method for nonlinear integral inequalities related to critical nonlinear wave equations
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