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

TL;DR

This paper develops a unified test function method to estimate the lifespan of solutions for semilinear parabolic, dispersive, and hyperbolic equations, revealing how domain geometry influences blowup thresholds.

Contribution

It introduces a unified approach to estimate solution lifespans across different equations and domains, extending classical results to cone-like geometries with explicit eigenvalue-based thresholds.

Findings

Threshold for blowup depends on domain geometry.

Fujita exponent remains a critical threshold in certain domains.

Explicit lifespan estimates are derived for cone-like domains.

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.