Time-weighted blow-up rates and pointwise profile for single-point blow-up solutions in reaction-diffusion equations

摘要

This paper deals with asymptotic behavior for blow-up solutions to time-weighted reaction-diffusion equations u(t) = Delta u+e(alpha t)v(p) and v(t) = Delta v+e(beta t)u(q), subject to homogeneous Dirichlet boundary. The time-weighted blow-up rates are defined and obtained by ways of the scaling or auxiliary-function methods for all alpha,beta epsilon R. Aiding by key inequalities between components of solutions, we give lower pointwise blow-up profiles for single-point blow-up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow-up rates and new blowup versus global existence criteria are obtained. Time-weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow-up rates, but they do not limit the order of time-weighted blow-up rates and pointwise profile near blow-up time. Copyright (C) 2017 JohnWiley & Sons, Ltd.