Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions

• Published in: Nonlinearity Vol. 31; no. 7; pp. 3228 - 3250
• Main Authors: Mingqi, Xiang, R dulescu, Vicen iu D, Zhang, Binlin
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.07.2018
• Subjects: blow-up; fractional Laplacian; Galerkin method; Kirchhoff-type diffusion problem; local existence
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/aaba35

In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.