Acta mathematica scientia,Series B ›› 2018, Vol. 38 ›› Issue (3): 843-856.doi: 10.1016/S0252-9602(18)30788-4

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BLOW-UP PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS ON LOCALLY FINITE GRAPHS

Yong LIN, Yiting WU   

  1. Department of Mathematics, Renmin University of China, Beijing 100872, China
  • Received:2017-04-14 Revised:2017-07-30 Online:2018-06-25 Published:2018-06-25
  • Contact: Yiting WU E-mail:yitingly@126.com
  • Supported by:

    The first author is supported by the National Science Foundation of China (11671401); the second author is supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (17XNH106).

Abstract:

Let G=(V, E) be a locally finite connected weighted graph, and △ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut=△u + f(u) on G. The blow-up phenomenons for ut=△u + f(u) are discussed in terms of two cases:(i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

Key words: Blow-up, parabolic equations, locally finite graphs, differential inequalities

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