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

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 ,...

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Published inNonlinearity Vol. 31; no. 7; pp. 3228 - 3250
Main Authors Mingqi, Xiang, R dulescu, Vicen iu D, Zhang, Binlin
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.07.2018
Subjects
blow-up
fractional Laplacian
Galerkin method
Kirchhoff-type diffusion problem
local existence
Online AccessGet full text
ISSN0951-7715
1361-6544
DOI10.1088/1361-6544/aaba35

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Abstract 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.
AbstractList 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.
Author R dulescu, Vicen iu D
Mingqi, Xiang
Zhang, Binlin
Author_xml – sequence: 1
  givenname: Xiang
  surname: Mingqi
  fullname: Mingqi, Xiang
  email: xiangmingqi_hit@163.com
  organization: Civil Aviation University of China College of Science, Tianjin 300300, People's Republic of China
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  givenname: Vicen iu D
  orcidid: 0000-0003-4615-5537
  surname: R dulescu
  fullname: R dulescu, Vicen iu D
  email: vicentiu.radulescu@imar.ro
  organization: University of Craiova Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania
– sequence: 3
  givenname: Binlin
  surname: Zhang
  fullname: Zhang, Binlin
  email: zhangbinlin2012@163.com
  organization: Heilongjiang Institute of Technology Department of Mathematics, Harbin 150050, People's Republic of China
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Snippet 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...
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SubjectTerms blow-up
fractional Laplacian
Galerkin method
Kirchhoff-type diffusion problem
local existence
Title Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions
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