Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations. (17th March 2017)

• Record Type: Journal Article
• Authors: Liu, Bingchen; Li, Fengjie
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 14(2017)
• Page Start: 5273
• Page End: 5285
• Publication Date: 2017-03-17
• Subjects: time‐weighted blow‐up rate -- single‐point blow‐up -- pointwise blow‐up profile -- nonstandard growth source; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4385
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations u t =Δ u + e α t v p and v t =Δ v + e β 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 α, β ∈ 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 blow‐up 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 © 2017 John Wiley & Sons, Ltd.