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

By Xiang Mingqi, Vicentiu D. Rădulescu, Binlin Zhang. Nonlinearity, 2018

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/3b198641c193719ffbaf0731cc72e765921bfec0

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In Applied Mathematics

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Citations in the corpus

Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity (2018, background): “For more details on the physical aspects, we refer the readers to [2, 3, 7, 8, 26].”