PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION

doi:10.1016/S0252-9602(14)60046-1

Abstract

Abstract:

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.

Key words: Nonlocal boundary condition, blow-up, blow-up rate, Porous medium equation

CLC Number:

35B35

LI Yu-Huan, MI Yong-Sheng, MU Chun-Lai. PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION[J].Acta mathematica scientia,Series B, 2014, 34(3): 748-758.

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References

[1] Galaktionov V A. On asymptotic self-similar behavior for a quasilinear heat equation: Single point blow-up. SIAM J Math Anal, 1995, 26: 675–693

[2] Samarskii A A, Kurdyumov S P, Galaktionov V A, Mikhailov A P. Blow-up in Problems for Quasilinear Parabolic Equations (in Russian). Moscow: Nauka, 1987; Berlin: Walter de Gruyter, 1995

[3] Cantrell R S, Cosner C. Diffusive logistic equations with indefinite weights: Population models in disrupted environments II. SIAM J Math Anal, 1989, 22: 1043–1064

[4] Alikakos N D, Rostamian R. Large time behavior of solutions of Neumann boundary value problem for the porous medium equation. Indiana Univ Math J, 1981, 30: 749–785

[5] Aronson D C. The porous medium equation//Lecture Notes in Math. 1224: Nonlinear Diffusion Problem. Berlin: Springer, 1986: 1–46

[6] Bebernes J, Galaktionov V A. On classification of blow-up patterns for a quasilinear heat equation. Diff Int Equs, 1996, 9: 655–670

[7] Wang Y, Mu C, Xiang Z. Blow up of solutions to a porous medium equation with nonlocal boundary condition. Appl Math Comput, 2007, 192: 579–585

[8] Levine H A. The role of critical exponents in blow-up theorems. SIAM Rev, 1990, 32: 262–288

[9] Li F, Xie C. Global existence and blow-up for a nonlinear porous medium equation. Appl Math Lett, 2003, 16: 185–192

[10] Wu Z Q, Zhao J N, Yin J X, Li H L. Nonlinear Diffusion Equations. Inc River Edge, NJ: World Scientific Publishing C, 2001

[11] Galaktionov V A, Levine H A. On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel J Math, 1996, 94: 125–146

[12] Galaktionov V G. Asymptotic behavior of unbounded solution of the nonlinear parabolic equation u_t=(uu_x)x + u1+. Diff Equs, 1985, 21: 751–759

[13] Galaktionov V G. Proof of the localization of unbounded solutions of the nonlinear parabolic equation u_t= (uu_x)x + u. Diff Equs, 1985, 21: 15–23

[14] Anderson J R, Deng K. Global existence for degenerate parabolic equations with non-local forcing. Math Methods Appl Sci, 1997, 20: 1069–1087

[15] Du L L, Mu C L, Fan M S. Global existence and nonexistence for a quasilinear degenerate parabolic system with nonlocal source. Dynamical Systems, 2005, 20: 401–412

[16] Deng W B, Li Y X, Xie C H. Blow-up and global existence for a nonlocal degenerate parabolic system. J Math Anal Appl, 2003, 277: 199–217

[17] Duan Z W, Deng W B, Xie C H. Uniform blow-up profile for a degenerate parabolic system with nonlocal source. Comput Math Appl, 2004, 47: 977–995

[18] Song X F, Zheng S N, Jiang Z X. Blow-up analysis for a nonlinear diffusion system. Z Angew Math Phys, 2005, 56: 1–10

[19] Day W A. A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quart Appl Math, 1983, 40: 468–475

[20] Day W A. Heat Conduction within Linear Thermoelasticity//Vol.30 of Springer Tracts in Natural Philos-ophy. New York, NY, USA: Springer, 1985

[21] Friedman A. Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quart Appl Math, 1986, 44: 401–407

[22] Seo S. Blowup of solutions to heat equations with nonlocal boundary conditions. Kobe J Math, 1996, 13: 123–132

[23] Seo S. Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions. Pacific J Math, 2000, 193: 219–226

[24] Deng K. Comparison principle for some nonlocal problems. Quart Appl Math, 1992, 50: 517–522

[25] Pao C V. Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary condi-tions. Comput Math Appl, 1998, 88: 225–238

[26] Gladkov A, Ik Kim K. Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition. J Math Anal Appl, 2008, 338: 264–273

[27] Kong L H, Wang M X. Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries. Sci China Series A, 2007, 50: 1251–1266

[28] Wang Y L, Xiang Z Y. Blowup Analysis for a Semilinear Parabolic System with Nonlocal Boundary Condition. Boundary Value Problems. Volume 2009, Article ID 516390

[29] Wang Y L, Mu C L, Xiang Z Y. Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary. Boundary Value Problems, 2007, Article ID 64579

[30] Yin H M. On a class of parabolic equations with nonlocal boundary conditions. J Math Anal Appl, 2004, 294: 712–728

[31] Yin Y F. On nonlinear parabolic equations with nonlocal boundary condition. J Math Anal Appl, 1994, 185: 161–174

[32] Zheng S N, Kong L. Roles of weight functions in a nonlinear nonlocal parabolic system. Nonlinear Anal, 2008, 68: 2406–2416

[33] Mu C L, Liu D M. Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions. Abstr Appl Anal, 2010, (2010), Article ID 148035

[34] Mu C L, Liu D M. positive solutions for the reaction-diffusion equation with nonlocal nonlinear boundary condition. J Korean Math Soc, 2010, 47: 1317–1328

[35] Anderson J R. Local existence and uniqueness of solutions of degenerate parabolic equations. Commun Partial Diff Eqns, 1991, 16: 105–143

[36] Deng K. Comparison principle for some nonlocal problems. Quart Appl Math, 1992, 50: 517–522

[37] Souplet P. Blow up in nonlocal reaction-diffusion equations. SIAM J Math Anal, 1998, 29: 1301–1334

[38] Friedman A, Mcleod J B. Blow-up of positive solutions of semilinear heat equations. Indiana Univ Math J, 1985, 34: 425–447