Exploding solutions for a nonlocal quadratic evolution problem

Abstract

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 -norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczyński [8].

Blowup of solutions to generalized Keller–Segel model

“…2] in order to show that u − (x, t) ≡ 0. Here, we do not give a detailed presentation because the proof from [24,Lemma 2.7] can be rewritten in this more general case. Thus, the nonnegativity property is natural for solutions of (1.1), and our results show that loss of regularity (blowup) and loss of nonnegativity are intimately connected.…”

“…13886SG, and by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge "Harmonic Analysis, Nonlinear Analysis and Probability" MTKD-CT-2004-013389. The authors are greatly indebted to Tomasz Cieślak for pointing them out the preprint [24] and to Takayoshi Ogawa for the reprint of [22].…”

“…Thus, we mean this phenomenon as blowup for (1.1). After this paper was completed, we discovered a recent preprint [24] where the authors show the blowup of solution to system (1.1)-(1.3) with fractional diffusion in the particular case d = 2 and β = 2. Our argument is different than that in [24], shorter, seems to be more direct, and applies in more general situations.…”

“…After this paper was completed, we discovered a recent preprint [24] where the authors show the blowup of solution to system (1.1)-(1.3) with fractional diffusion in the particular case d = 2 and β = 2. Our argument is different than that in [24], shorter, seems to be more direct, and applies in more general situations. Moreover, we are able to formulate a simple condition on the initial data which leads to the blowup in a finite time of the corresponding solution.…”

On the generalized Buckley-Leverett equation

“…However, we remark that it can also be obtained by means of pointwise arguments (see Castro & Córdoba [7] for an application of these pointwise arguments to prove blow up). These virial-type arguments have been used for several transport equations even in the case of nonlocal velocities (see Córdoba,Córdoba & Fontelos [12], Dong, Du & Li [14], Li & Rodrigo [28] and Li, Rodrigo & Zhang [29]). In this case, the transport term is highly nonlinear and this method fails in the case of viscosity ν > 0, 0 < α ≪ 1.…”