Upper bound for lifespan of solutions to certain semilinear parabolic, dispersive and hyperbolic equations via a unified test function method

Abstract

This paper is concerned with the blowup phenomena for initial-boundary value problem for certain semi linear parabolic, dispersive and hyperbolic equations in cone-like domain. The result proposes a unified treatment of estimates for lifespan of solutions to the problem by test function method. The Fujita exponent p=1 + 2/N appears as a threshold of blowup phenomena for small data when $C_{\Sigma }=R^N$ , but the case of cone-like domain with boundary the threshold changes and explicitly given via the first eigenvalue of corresponding Laplace-Beltrami operator with Dirichlet boundary condition as in Levine-Meier in 1989.

We prove the sharp upper bound of lifespan to semilinear heat equation, damped wave equation and Schrödinger equation in the Euclidean space in L^1-setting for convenience of readers

References in corpus (3)

• On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
• Life-span of blowup solutions to semilinear wave equation with space-dependent critical damping
• A remark on the critical exponent for the semilinear damped wave equation on the half-space

Cited by in corpus (4)

• Test function method for blow-up phenomena of semilinear wave equations and their weakly coupled systems
• Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities
• A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms
• Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain