Grow-up rate of solutions for the heat equation with a sublinear source
© Wang and Yin; licensee Springer 2012
Received: 13 June 2012
Accepted: 20 August 2012
Published: 31 August 2012
In this paper, we investigate the grow-up rate of solutions for the heat equation with a sublinear source. We find that if the initial value grows fast enough, then it plays a major role in the growing up of solutions, while if the initial value grows slowly, then the sublinear source prevails. As a direct application of these results, we show that the effect of the sublinear source is negligible in the asymptotic behavior of solutions as if the initial value grows fast enough.
Keywordsgrow-up asymptotic behavior heat equation sublinear source
Here , , and with .
see also . For the Cauchy-Dirichlet problem of (1.1), the existence of growing up global solutions and the grow-up rate of solutions has been investigated by Dold, Galaktionov, Lacey and Vázquez in , Galaktionov and King in . If , there are also a lot of papers which intensely investigate the solutions of (1.1)-(1.2) converging to zero at different algebraic rates [11–16].
then the solutions of (1.1)-(1.2) are global.
for large t. Here if , and for any if . Moreover, as an application of these results, we get that if and the initial value satisfies (1.3), (1.4), then the effect of the sublinear source is negligible in the asymptotic behavior of solutions as . While for , Aguirre and Escobedo  revealed that the effect of the sublinear source cannot be negligible in the asymptotic behavior of the solutions as . For the absorption case ( is replaced by in (1.1)) and the supercritical case ( in (1.1)), some similar results about the asymptotic behavior of solutions for these problems were established by a lot of papers, see [18–20].
The paper is organized as follows. The next section is devoted to giving the grow-up rate for the solutions of the problem (1.1)-(1.2) with . In Section 3, we investigate the asymptotic behavior of solutions for the problem (1.1)-(1.2).
2 Growth-up rate of solutions
So, for any , those two growing up effects given by (2.1) and (2.2) can be compared as . When , the one given by (2.2) prevails; when , the one given by (2.1) prevails; and they coincide in the critical case .
If the initial value , the existence and uniqueness of a mild solution for the problem (1.1)-(1.2) has been given in .
Lemma 2.1 ()
for a.e. .
Moreover, if , the convergence is uniform on compact subsets of .
Our results about the grow-up rate of solutions are the following two theorems.
Here is the solution of (1.1)-(1.2).
From this, we get (2.7) easily. So we complete the proof of this theorem. □
In the following theorem, we consider the grow-up rate for the solutions of the problem (1.1)-(1.2) when the nonnegative initial value with .
Here is also the solution of (1.1)-(1.2).
Combining this with (2.14), we can get the desired results. So we complete the proof of this theorem. □
Remark 2.1 From Theorem 2.1 and Theorem 2.2, we find that if , then the main effect on the growing up of solutions comes from the initial value; while if , then the sublinear source has a major effect.
3 Asymptotic behavior
for any which vanishes for large and at .
The following result gives the fact that if , then the sublinear source is negligible in the asymptotic behavior of the rescaled solution as . Similar to [19, 22, 23], we follow the framework by Kamin and Peletier  to give the proof of our result.
uniformly on sets , . Here is the mild solution of (1.1)-(1.2) and .
uniformly on any compact sets of as . By (3.8), we thus have (3.2). So we complete the proof of this theorem. □
By (A.1) and the comparison principle, we have (A.2). So we complete the proof of this lemma. □
The authors are grateful to the referees for their useful comments. This work are supported by NSFC, the Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of “CQ CSTC”, the Scientific and Technological Projects of Chongqing Municipal Commission of Education.
- Fujita H:On the blowing up of solutions of the Cauchy problem for . J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 1966, 13: 109-124.Google Scholar
- Hayakawa K: On the nonexistence of global solutions of some semilinear parabolic equations. Proc. Jpn. Acad. 1974, 49: 503-525.MathSciNetView ArticleGoogle Scholar
- Galaktionov VA: Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. R. Soc. Edinb. A 1994, 124(3):517-525. English summary 10.1017/S0308210500028766MathSciNetView ArticleGoogle Scholar
- Lee T-Y, Ni W-M: Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 1992, 333(1):365-378. 10.1090/S0002-9947-1992-1057781-6MathSciNetView ArticleGoogle Scholar
- Poláčik P, Yanagida E: On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 2003, 327(4):745-771. English summary 10.1007/s00208-003-0469-yMathSciNetView ArticleGoogle Scholar
- Poláčik P, Yanagida E: Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation. Differ. Integral Equ. 2004, 17(5-6):535-548. English summaryGoogle Scholar
- Fila M, Winkler M, Yanagida E: Grow-up rate of solutions for a supercritical semilinear diffusion equation. J. Differ. Equ. 2004, 205(2):365-389. English summary 10.1016/j.jde.2004.03.009MathSciNetView ArticleGoogle Scholar
- Mizoguchi N: Grow-up of solutions for a semilinear heat equation with supercritical nonlinearity. J. Differ. Equ. 2006, 227(2):652-669. English summary 10.1016/j.jde.2005.11.002MathSciNetView ArticleGoogle Scholar
- Dold JW, Galaktionov VA, Lacey AA, Vázquez JL: Rate of approach to a singular steady state in quasilinear reaction-diffusion equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 1998, 26(4):663-687.Google Scholar
- Galaktionov VA, King JR: Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents. J. Differ. Equ. 2003, 189(1):199-233. English summary 10.1016/S0022-0396(02)00151-1MathSciNetView ArticleGoogle Scholar
- Galaktionov V, Vázquez JL: Continuation of blow-up solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 1997, 50: 1-67. 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-HView ArticleGoogle Scholar
- Gui C, Ni W-M, Wang X: Further study on a nonlinear heat equation. J. Differ. Equ. 2001, 169(2):588-613. 10.1006/jdeq.2000.3909MathSciNetView ArticleGoogle Scholar
- Fila M, King JR, Winkler M, Yanagida E: Linear behaviour of solutions of a superlinear heat equation. J. Math. Anal. Appl. 2008, 340(1):401-409. English summary 10.1016/j.jmaa.2007.08.029MathSciNetView ArticleGoogle Scholar
- Stinner C: Very slow convergence to zero for a supercritical semilinear parabolic equation. Adv. Differ. Equ. 2009, 14(11-12):1085-1106. English summaryMathSciNetGoogle Scholar
- Fila M, Winkler M, Yanagida E: Slow convergence to zero for a parabolic equation with a supercritical nonlinearity. Math. Ann. 2008, 340(3):477-496. English summaryMathSciNetView ArticleGoogle Scholar
- Stinner C: Very slow convergence rates in a semilinear parabolic equation. Nonlinear Differ. Equ. Appl. 2010, 17(2):213-227. English summary 10.1007/s00030-009-0050-9MathSciNetView ArticleGoogle Scholar
- Aguirre J, Escobedo MA:Cauchy problem for with . Asymptotic behaviour of solutions. Ann. Fac. Sci. Toulouse Math. (5) 1986/87, 8(2):175-203.MathSciNetView ArticleGoogle Scholar
- Galaktionov VA, Vázquez JL: Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach. J. Funct. Anal. 1991, 100(2):435-462. 10.1016/0022-1236(91)90120-TMathSciNetView ArticleGoogle Scholar
- Kamin S, Peletier LA: Large time behaviour of solutions of the heat equation with absorption. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 1985, 12(3):393-408.MathSciNetGoogle Scholar
- Ishige K, Ishiwata M, Kawakami T: The decay of the solutions for the heat equation with a potential. Indiana Univ. Math. J. 2009, 58(6):2673-2707. English summary 10.1512/iumj.2009.58.3771MathSciNetView ArticleGoogle Scholar
- Cazenave T, Dickstein F, Escobedo M, Weissler FB: Self-similar solutions of a nonlinear heat equation. J. Math. Sci. Univ. Tokyo 2001, 8(3):501-540. English summaryMathSciNetGoogle Scholar
- Kamin S, Peletier LA: Large time behaviour of solutions of the porous media equation with absorption. Isr. J. Math. 1986, 55(2):129-146. 10.1007/BF02801989MathSciNetView ArticleGoogle Scholar
- Kwak M: A porous media equation with absorption. I. Long time behaviour. J. Math. Anal. Appl. 1998, 223(1):96-110. 10.1006/jmaa.1998.5961MathSciNetView ArticleGoogle Scholar
- Widder DV Pure and Applied Mathematics 67. In The Heat Equation. Academic Press, New York; 1975.Google Scholar
- Wu Z, Yin J, Wang C: Elliptic and Parabolic Equations. World Scientific, Hackensack; 2006.View ArticleGoogle Scholar
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