Open Access
March, 2010 Exploding solutions for a nonlocal quadratic evolution problem
Dong Li , José L. Rodrigo , Xiaoyi Zhang
Rev. Mat. Iberoamericana 26(1): 295-332 (March, 2010).


We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the $L_x^\infty$-norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczy\'nski [Biler, P. and Woyczy\'Nski, W.A.: Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math. {\bf 59} (1999), no. 3, 845-869].


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Dong Li . José L. Rodrigo . Xiaoyi Zhang . "Exploding solutions for a nonlocal quadratic evolution problem." Rev. Mat. Iberoamericana 26 (1) 295 - 332, March, 2010.


Published: March, 2010
First available in Project Euclid: 16 February 2010

zbMATH: 1195.35182
MathSciNet: MR2666316

Primary: 35A05 , 35A07 , 35K55 , 35K57

Keywords: chemotaxis , Fractional diffusion , nonlinear parabolic equation

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.26 • No. 1 • March, 2010
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