Revista Matemática Iberoamericana

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Volume 26, Issue 1, 2010, pp. 295–332
DOI: 10.4171/RMI/602

Published online: 2010-04-30

Exploding solutions for a nonlocal quadratic evolution problem

Dong Li[1], José L. Rodrigo[2] and Xiaoyi Zhang[3]

(1) University of Iowa, Iowa City, United States
(2) University of Warwick, Coventry, UK
(3) University of Iowa, Iowa City, USA

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].

Keywords: Nonlinear parabolic equation, fractional diffusion, chemotaxis

Li Dong, Rodrigo José, Zhang Xiaoyi: Exploding solutions for a nonlocal quadratic evolution problem. Rev. Mat. Iberoamericana 26 (2010), 295-332. doi: 10.4171/RMI/602