Improved conditions for single-point blow-up in reaction-diffusion systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Mahrnoudi, Nejib¹  Souplet, Philippe²  Tayachi, Slim¹ 

[1] Univ Tunis El Manan, Fac Sci Tunis, Dept Math, Lab Equat Derivees Partielles LR03ES04, Tunis 2092, Tunisia

[2] Univ Paris 13, CNRS UMR 7539, Sorbonne Paris Cite, Lab Anal Geometrie & Applicat, F-93430 Villetaneuse, France

关键词: Nonlinear initial boundary value problems;    Nonlinear parabolic equations;    Reaction diffusion systems;    Asymptotic behavior of solutions;    Single-point blow-up;    Blow-up profile;   

DOI  :  10.1016/j.jde.2015.03.024

来源: Elsevier

【 摘 要 】

We study positive blowing-up solutions of the system: u(t)-delta Delta u = v(P), v(t)-Delta v = u(q) , as well as of some more general systems. For any p, q> 1, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions: (i) no type I blow-up assumption is made (and it is known that this property may fail); (ii) no equidiffusivity is assumed, i.e. any delta > 0 is allowed; (iii) a large class of nonlinearities F(u, v), G(u, v) can be handled, which need not follow a precise power behavior. As a side result, we also obtain lower pointwise estimates for the final blow-up profiles. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free