BLOW-UP CONDITIONS FOR A SEMILINEAR PARABOLIC SYSTEM ON LOCALLY FINITE GRAPHS

doi:10.1007/s10473-024-0213-0

Abstract

Abstract: In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition $CDE'(n,0)$, the polynomial volume growth of degree $m$, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).

Key words: semilinear parabolic system on graphs, blow-up, heat kernel estimate on graphs

CLC Number:

35R02

Yiting WU. BLOW-UP CONDITIONS FOR A SEMILINEAR PARABOLIC SYSTEM ON LOCALLY FINITE GRAPHS[J].Acta mathematica scientia,Series B, 2024, 44(2): 609-631.

Trendmd

References

[1] Andreucci D, Herrero M A, Velázquez J J L. Liouville theorems and blow up behaviour in semilinear reaction diffusion systems. Ann Inst Henri Poincar 'e Anal Non Lin 'eaire, 1997, 14(1): 1-53
[2] Escobedo M, Herrero M A. Boundedness and blow up for a semilinear reaction-diffusion system. J Diff Equations, 1991, 89: 176-202
[3] Escobedo M, Herrero M A. A semilinear parabolic system in a bounded domain. Annali Mat Pura Appl, 1993, 165(4): 315-336
[4] Ge H. Kazdan-Warner equation on graph in the negative case. J Math Anal Appl, 2017, 453(2): 1022-1027
[5] Ge H. The $p$th Kazdan-Warner equation on graphs. Commun Contemp Math, 2020, 22(6): 1950052
[6] Ge H, Hua B, Jiang W. A note on Liouville equations on graphs. Proc Amer Math Soc, 2018, 146(11): 4837-4842
[7] Grigor'yan A, Lin Y, Yang Y. Kazdan-Warner equation on graph. Calc Var Partial Differential Equations, 2016, 55(4): Art 92
[8] 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
[9] Horn P, Lin Y, Liu S, Yau S T. Volume doubling, Poincar 'e inequality and Gaussian heat kernel estimate for non-negatively curved graphs. J Reine Angew Math, 2019, 757: 89-130
[10] Hua B, Li R. The existence of extremal functions for discrete Sobolev inequalities on lattice graphs. J Diff Equations, 2021, 305: 224-241
[11] Hua B, Mugnolo D. Time regularity and long-time behavior of parabolic $p$-Laplace equations on infinite graphs. J Diff Equations, 2015, 259(11): 6162-6190
[12] Huang X. On uniqueness class for a heat equation on graphs. J Math Anal Appl, 2012, 393(2): 377-388
[13] Keller M, Lenz D. Dirichlet forms and stochastic completeness of graphs and subgraphs. J Reine Angew Math, 2012, 666: 189-223
[14] 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): Art 102
[15] Lin Y, Wu Y. Blow-up problems for nonlinear parabolic equations on locally finite graphs. Acta Math Sci, 2018, 38B(3): 843-856
[16] Lin Y, Xie Y. Application of Rothe's method to a nonlinear wave equation on graphs. B Korean Math Soc, 2022, 59(3): 745-756
[17] Lin Y, Yang Y. A heat flow for the mean field equation on a finite graph. Calc Var Partial Differential Equations, 2021, 60(6): Art 206
[18] Liu S, Yang Y. Multiple solutions of Kazdan-Warner equation on graphs in the negative case. Calc Var Partial Differential Equations, 2020, 59(5): Art 164
[19] Weber A. Analysis of the physical Laplacian and the heat flow on a locally finite graph. J Math Anal Appl, 2010, 370(1): 146-158
[20] Wojciechowski R. Heat kernel and essential spectrum of infinite graphs. Indiana Univ Math J, 2009, 58(3): 1419-1442
[21] Wu Y. Blow-up for a semilinear heat equation with Fujita's critical exponent on locally finite graphs. Rev R Acad Cien Serie A Mat, 2021, 115(3): Art 133
[22] Zhang N, Zhao L. Convergence of ground state solutions for nonlinear Schrodinger equations on graphs. Sci China Math, 2018, 61(8): 1481-1494