Nonexistence for parabolic differential inequalities with convection terms in exterior domains

TL;DR

This paper establishes criteria for the nonexistence of global weak solutions to semilinear parabolic inequalities with convection in exterior domains, considering weighted nonlinearities and different boundary conditions, extending previous Fujita-type results.

Contribution

It introduces a unified approach to determine nonexistence criteria for solutions with weighted nonlinearities and boundary conditions, improving upon previous Fujita exponent results.

Findings

Derived sufficient conditions for nonexistence of solutions.

Identified a larger Fujita-type critical exponent when b5=0.

Extended nonexistence results to exterior domains with convection terms.

Abstract

We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^\alpha |x|^\sigma$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $\alpha=0$, the critical exponent in the sense of Fujita is obtained. This exponent is bigger than that found previously by Zheng and Wang (2008) in the case of homogeneous Neumann and Dirichlet boundary conditions.