LARGE TIME BEHAVIOR OF SOLUTIONS TO NEWTONIAN FILTRATION EQUATIONS WITH SOURCES

doi:10.1016/S0252-9602(10)60094-X

Abstract

Abstract:

This article is concerned with large time behavior of solutions to the Neumann or Dirichlet problem for a class of Newtonian filtration
equations

|x|^λ+k ∂u / ∂t =div(|x|^k\nabla u^m)+|x|^λ+ku^p

with 0<m<1, p>1, λ ≥0, k ∈ R. An interesting phenomenon is that there exist two thresholds k_∞ and k₁ for the exponent k, such that the critical Fujita exponent p_c for p exists and is finite if k ∈ (k_∞, k₁), otherwise, p_c is infinite or does not exist.

Key words: Newtonian filtration equation, critical exponent, exterior problem, Threshold

CLC Number:

35B33

WANG Lu-Sheng, YIN Jing-Xue, WANG Ze-Jia. LARGE TIME BEHAVIOR OF SOLUTIONS TO NEWTONIAN FILTRATION EQUATIONS WITH SOURCES[J].Acta mathematica scientia,Series B, 2010, 30(3): 968-974.

Trendmd

References

[1] Kalashnikov A S. Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations. Russian
Math Surveys, 1987, 42(2): 169--222

[2] Wu Z Q. Degenerate quasilinear parabolic equations (in Chinese). Adv in Math (Beijing), 1987, 16(2): 121--158

[3] 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 I, 1996, 13: 109--124

[4] Deng K, Levine H A. The role of critical exponents in blow-up theorems: the sequel. J Math Anal Appl, 2000, 243(1): 85--126

[5] Levine H A. The role of critical exponents in blow-up theorems. SIAM Rev, 1990, 32(2): 262--288

[6] Ackleh A S, Deng K. On the critical exponent for the Schrodinger equation with a nonlinear boundary condition. Differential Integral Equations, 2004, 17(11/12): 1293--1307

[7] Ikehata R. Critical exponent for semilinear damped wave equations in the N-dimensional half space. J Math Anal Appl, 2003, 288(2): 803--818

[8] Quiros F, Rossi J D. Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana Univ Math J, 2001, 50: 629--654

[9] Wang C P, Zheng S N. Critical Fujita exponents of degenerate and singular parabolic equations. Proc Roy Soc Edinburgh Sect A, 2006, 136: 415--430

[10] Wang Z J, Yin J X, Wang L S. Critical exponent for non-Newtonian filtration equation with homogeneous Neumann boundary data. Math Methods Appl Sci, 2007, in press

[11] Winkler M. A critical exponent in a degenerate parabolic equation. Math Methods Appl Sci, 2002, 25: 911--925

[12] Lady\vzenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and quasilinear equations of parabolic type. Transl Math Mono, 23, AMS. Providence RI, 1967

[13] Wu Z Q, Zhao J N, Yin J X et al. Nonlinear diffusion equations. Singapore: World Scientific Publishing Co, Inc, 2001

[14] Qi Y W. The critical exponents of parabolic equations and blow-up in ${\mathbb R}^n$. Proc Roy Soc Edinburgh Sect A, 1998, 128(1): 123--136

[1]	Lingyan YANG, Xiaoguang LI, Yonghong WU, Louis CACCETTA. GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION [J]. Acta mathematica scientia,Series B, 2017, 37(4): 941-948.
[2]	Kamal OULD BOUH. EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY [J]. Acta mathematica scientia,Series B, 2016, 36(5): 1305-1316.
[3]	Jing ZHANG, Shiwang MA. INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BRÉZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL [J]. Acta mathematica scientia,Series B, 2016, 36(2): 527-536.
[4]	Pan ZHENG, Chunlai MU, Xuegang HU, Fuchen ZHANG. SECONDARY CRITICAL EXPONENT AND LIFE SPAN FOR A DOUBLY SINGULAR PARABOLIC EQUATION WITH A WEIGHTED SOURCE [J]. Acta mathematica scientia,Series B, 2016, 36(1): 244-256.
[5]	Haixia LI, Yuzhu HAN, Wenjie GAO. CRITICAL EXTINCTION EXPONENTS FOR POLYTROPIC FILTRATION EQUATIONS WITH NONLOCAL SOURCE AND ABSORPTION [J]. Acta mathematica scientia,Series B, 2015, 35(2): 366-374.
[6]	WANG Li, WANG Ji-Xiu. CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS [J]. Acta mathematica scientia,Series B, 2014, 34(5): 1603-1618.
[7]	SHANG Yue-Yun, WANG Li. MULTIPLE NONTRIVIAL SOLUTIONS FOR A CLASS OF SEMILINEAR POLYHARMONIC EQUATIONS [J]. Acta mathematica scientia,Series B, 2014, 34(5): 1495-1509.
[8]	SUN Xiao-Mei. p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL [J]. Acta mathematica scientia,Series B, 2013, 33(4): 1099-1112.
[9]	LIU Xiao-Chun, MEI Yuan. EXISTENCE OF NODAL SOLUTION FOR SEMI-LINEAR ELLIPTIC EQUATIONS WITH CRITICAL SOBOLEV EXPONENT ON SINGULAR#br# MANIFOLD [J]. Acta mathematica scientia,Series B, 2013, 33(2): 543-555.
[10]	DING Ling, TANG Chun-Lei. POSITIVE SOLUTIONS FOR CRITICAL QUASILINEAR ELLIPTIC EQUATIONS WITH MIXED DIRICHLET-NEUMANN BOUNDARY CONDITIONS [J]. Acta mathematica scientia,Series B, 2013, 33(2): 443-470.
[11]	LING Zheng-Qiu, WANG Ze-Jia, ZHANG Guo-Qiang. THE SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP CRITERIA FOR A DIFFUSION SYSTEM [J]. Acta mathematica scientia,Series B, 2013, 33(1): 139-149.
[12]	YIN Jing-Hua, JIN Chun-Hua, YANG Ying. CRITICAL EXPONENTS OF EVOLUTIONARY p-LAPLACIAN WITH INTERIOR AND BOUNDARY SOURCES [J]. Acta mathematica scientia,Series B, 2011, 31(3): 778-790.
[13]	KANG Dong-Sheng. SOLUTIONS FOR THE QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY--SOBOLEV EXPONENTS [J]. Acta mathematica scientia,Series B, 2010, 30(5): 1529-1540.
[14]	ZENG Xian-Zhong, LIU Zhen-Hai. EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC#br# EQUATIONS IN EXTERIOR DOMAINS [J]. Acta mathematica scientia,Series B, 2010, 30(3): 713-725.
[15]	LI Gong-Bao, ZHANG Guo. MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT [J]. Acta mathematica scientia,Series B, 2009, 29(4): 903-918.