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

Yong LIN; Yiting WU

doi:10.1016/S0252-9602(18)30788-4

Acta mathematica scientia, Series B >

2018 , Vol. 38 >Issue 3: 843 - 856

DOI: https://doi.org/10.1016/S0252-9602(18)30788-4

Articles

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

Yong LIN ,
Yiting WU

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Department of Mathematics, Renmin University of China, Beijing 100872, China

Received date: 2017-04-14

Revised date: 2017-07-30

Online published: 2018-06-25

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).

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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 u_t=△u + f(u) on G. The blow-up phenomenons for u_t=△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

Cite this article

Yong LIN , Yiting WU . BLOW-UP PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS ON LOCALLY FINITE GRAPHS[J]. Acta mathematica scientia, Series B, 2018 , 38(3) : 843 -856 . DOI: 10.1016/S0252-9602(18)30788-4

References

[1] Bauer F, Hua B, Jost J. The dual cheeger constant and spectra of infinite graphs. Adv Math, 2014, 251(2):147-194
[2] Bebernes J, Eberly D. Mathematical Problems from Combustion Theory. New York:Springer, 1989
[3] Chung F R K. Spectral Graph Theory, CBMS Reg Conf Ser Math. Providence, RI:Amer Math Soc, 1997
[4] Chung Y S, Lee Y S, Chung S Y. Extinction and positivity of the solutions of the heat equations with absorption on networks. J Math Anal Appl, 2011, 380(2):642-652
[5] Dodziuk J, Kendall W S. Combinatorial Laplacians and the isoperimetric inequality//Ellworthy K D. From Local Times to Global Geometry. Control and Physics. Pitman Research Notes in Mathematics Series 150. Essex:Longman Scientific and Technical, 1986:68-74
[6] Fujita H. On the blowing up of solutions of the Cauchy problem for u_t=△u+u^1+α. J Fac Sci Univ Tokyo Sect A Math, 1966, 13(2):109-124
[7] Fujita H. On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. Symp Pure Mathematics, 1969, 18:105-113
[8] Grigor'yan A. Analysis on Graphs. University Bielefeld, 2009
[9] Grigor'yan A, Lin Y, Yang Y. Yamabe type equations on graphs. J Differential Equations, 2016, 261(9):4924-4943
[10] Grigor'yan A, Lin Y, Yang Y. Kazdan-Warner equation on graph. Calc Var Partial Differential Equations, 2016, 55(4):92(13 pages)
[11] Grigor'yan A, Lin Y, Yang Y. Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci China Math, 2017, 60(7):1311-1324
[12] Haeseler S, Keller M, Lenz D, Wojciechowski R. Laplacians on infinite graphs:Dirichlet and Neumann boundary conditions. J Spectr Theory, 2012, 2(4):397-432
[13] Kaplan S. On the growth of solutions of quasilinear parabolic equations. Comm Pure Appl Math, 1963, 16:305-333
[14] Lin Y, Wu Y. On-diagonal lower estimate of heat kernels on graphs. J Math Anal Appl, 2017, 456:1040-1048
[15] Lin Y, Wu Y. The existence and nonexistence of global solutions for a semilinear heat equation on graphs. Calc Var Partial Differential Equations, 2017, 56(4):102(22 pages)
[16] Liu W, Chen K, Yu J. Extinction and asymptotic behavior of solutions for the ω-heat equation on graphs with source and interior absorption. J Math Anal Appl, 2016, 435(1):112-132
[17] Osgood W F. Beweis der Existenz einer Lösung der Differentialgleichung dy/dx=f(x, y) ohne Hinzunahme der Cauchy-Lipschitzschen Bedingung. Monatshefte der Mathemratik unid Physik, 1898, 9(1):331-345
[18] Weber A. Analysis of the physical Laplacian and the heat flow on a locally finite graph. J Math Anal Appl, 2012, 370(1):146-158
[19] Wojciechowski R. Heat kernel and essential spectrum of infinite graphs. Indiana Univ Math J, 2008, 58(3):1419-1442
[20] Xin Q, Xu L, Mu C. Blow-up for the ω-heat equation with Dirichelet boundary conditions and a reaction term on graphs. Appl Anal, 2014, 93(8):1691-1701

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