Blow-up analysis for parabolic equations and systems via an auxiliary functional approach

Abstract

This paper considers the finite-time blow-up of solutions to semilinear parabolic equations and systems. Using a direct functional method, we first study single equations with source terms f ( u ) = u ^p and f ( u ) = u ^p + u ^q . We establish blow-up criteria, derive explicit upper bounds for the blow-up time, and obtain a new sufficient condition for blow-up. We then extend this method to coupled systems with bivariate power-type source terms. The construction of an auxiliary functional to show blow-up requires a modest assumption on the exponents, thereby providing alternative blow-up criteria to (X. Bai, Finite time blow-up for a reaction-diffusion system in bounded domain , Z. Angew. Math. Phys. 65 (2014), no. 1, 135–138, X. Wang, Global existence and finite time blow up for a reaction-diffusion system , Z. Angew. Math. Phys. 51 (2000), no. 1, 160–167, R. Xu, Y. Chen, Y. Yang, S. Chen, J. Shen, T. Yu, and Z. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrödinger equations , Electron. J. Differential Equations (2018), 52). Numerical evidence indicates that, in certain special domains, our estimates cannot be improved.