Complicated asymptotic behavior of solutions for a porous medium equation with nonlinear sources
© Wang and Yin; licensee Springer. 2013
Received: 12 July 2012
Accepted: 4 February 2013
Published: 21 February 2013
In this paper, we investigate the complicated asymptotic behavior of the solutions to the Cauchy problem of a porous medium equation with nonlinear sources when the initial value belongs to a weighted space.
AMS Subject Classification:35K55, 35B40.
where and with .
So, we can get our results by following the framework in  and using (1.6)-(1.7).
The rest of this paper is organized as follows. The next section is devoted to giving a sufficient condition for the global existence of solutions for problem (1.1)-(1.2) and the upper bounded estimates on these solutions. In the last section, we investigate the complicated asymptotic behavior of solutions.
2 Preliminaries and estimates
In this section we state the definition of a weak solution of problem (1.1)-(1.2) and give the upper bounded estimates on the global solutions. We begin with the definition of the weak solution of problem (1.1)-(1.2).
in and for each .
- 2.For and any nonnegative which vanishes for large , the following equation holds:(2.1)
A supersolution [or subsolution] is similarly defined with equality of (2.1) replaced by ≥ [or ≤]. The weak solutions for problem (1.4)-(1.5) can be defined in a similar way as above. It is well known that problem (1.1)-(1.2) has a unique, nonnegative and bounded weak solution, at least locally in time [16, 17]. Now we state the comparison principle for problem (1.1)-(1.2).
Hence they are both Banach spaces. The existence and uniqueness of a weak solution of problem (1.4)-(1.5) with the initial-value is shown in [16, 17], and this solution satisfies the following proposition.
Proposition 2.1 
In other words, . Moreover, if , then the semigroup is a contraction.
In the rest of this section, we give a sufficient condition for the existence of global solutions of problem (1.1)-(1.2) and establish the upper bounded estimates of these solutions.
where is a constant dependent only on M and η.
Remark 2.1 Notice that if and , then . So, our results capture Theorem 3 in . Here we use some ideas of them.
From this and (2.14), we can get (2.3). So, we complete the proof of this theorem. □
3 Complicated asymptotic behavior
In the rest of this section, we show that the complexity may occur in the asymptotic behavior of solutions of problem (1.1)-(1.2) with . Our main result is the following theorem.
uniformly on . Here is the solution of problem (1.1)-(1.2).
To get this theorem, we need to prove the following lemma first.
So, we complete the proof of this lemma. □
Now we can prove our main result.
Proof of Theorem 3.1
- I.For any , there exists a subsequence of the sequence satisfying
- II.There exists a constant satisfying
uniformly on . So, we complete the proof of Theorem 3.1. □
This work is supported by NSFC, the Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of ‘CQ CSTC’ (cstc2012jjA00013), the Scientific and Technological Projects of Chongqing Municipal Commission of Education (KJ121105).
- Galaktionov VA, Kurdjumov SP, Mihaĭov AP, Samarskiĭ AA:On unbounded solutions of the Cauchy problem for the parabolic equation . Sov. Math. Dokl. 1980, 252(6):1362-1364. (Russian)Google 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/S0308210500028766MATHMathSciNetView ArticleGoogle Scholar
- Kawanago T:Existence and behaviour of solutions for . Adv. Math. Sci. Appl. 1997, 7(1):367-400. (English summary)MATHMathSciNetGoogle Scholar
- Mochizuki K, Suzuki R: Critical exponent and critical blow-up for quasilinear parabolic equations. Isr. J. Math. 1997, 98: 141-156. (English summary) 10.1007/BF02937331MATHMathSciNetView ArticleGoogle Scholar
- Suzuki R: Asymptotic behavior of solutions of quasilinear parabolic equations with slowly decaying initial data. Adv. Math. Sci. Appl. 1999, 9(1):291-317. (English summary)MATHMathSciNetGoogle Scholar
- Mukai K, Mochizuki K, Huang Q: Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values. Nonlinear Anal. 2000, 39(1):33-45. 10.1016/S0362-546X(98)00161-8MATHMathSciNetView ArticleGoogle Scholar
- Suzuki R: Asymptotic behavior of solutions of quasilinear parabolic equations with supercritical nonlinearity. J. Differ. Equ. 2003, 190(1):150-181. (English summary) 10.1016/S0022-0396(02)00086-4MATHView ArticleGoogle Scholar
- Vázquez JL, Zuazua E: Complexity of large time behaviour of evolution equations with bounded data. Chin. Ann. Math., Ser. B 2002, 23(2):293-310. 10.1142/S0252959902000274MATHView ArticleGoogle Scholar
- Yin J, Liangwei W, Huang R: Complexity of asymptotic behavior of solutions for the porous medium equation with absorption. Acta Math. Sci. 2010, 30(6):1865-1880.MATHMathSciNetView ArticleGoogle Scholar
- Cazenave T, Dickstein F, Weissler FB:Universal solutions of the heat equation on . Discrete Contin. Dyn. Syst. 2003, 9(5):1105-1132. (English summary)MATHMathSciNetView ArticleGoogle Scholar
- Cazenave T, Dickstein F, Weissler FB:Universal solutions of a nonlinear heat equation on . Ann. Sc. Norm. Super. Pisa, Cl. Sci. 2003, 2(1):77-117. (English summary)MATHMathSciNetGoogle Scholar
- Cazenave T, Dickstein F, Weissler FB:Chaotic behavior of solutions of the Navier-Stokes system in . Adv. Differ. Equ. 2005, 10(4):361-398.MathSciNetGoogle Scholar
- Cazenave T, Dickstein F, Weissler FB:Nonparabolic asymptotic limits of solutions of the heat equation on . J. Dyn. Differ. Equ. 2007, 19(3):789-818. 10.1007/s10884-007-9076-zMATHMathSciNetView ArticleGoogle Scholar
- Carrillo JA, Vázquez JL: Asymptotic complexity in filtration equations. J. Evol. Equ. 2007, 7(3):471-495. (English summary) 10.1007/s00028-006-0298-zMATHMathSciNetView ArticleGoogle Scholar
- Kamin S, Peletier LA: Large time behaviour of solutions of the heat equation with absorption. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1985, 12(3):393-408.MATHMathSciNetGoogle Scholar
- Bénilan P, Crandall MG, Pierre M:Solutions of the porous medium equation in under optimal conditions on initial values. Indiana Univ. Math. J. 1984, 33(1):51-87. 10.1512/iumj.1984.33.33003MATHMathSciNetView ArticleGoogle Scholar
- Vázquez JL Oxford Mathematical Monographs. In The Porous Medium Equation. Mathematical Theory. Clarendon, Oxford; 2007.Google Scholar
- DiBenedetto E, Herrero MA: On the Cauchy problem and initial traces for a degenerate parabolic equation. Trans. Am. Math. Soc. 1989, 314(1):187-224.MATHMathSciNetView ArticleGoogle Scholar
- Yin J, Wang L, Huang R:Complexity of asymptotic behavior of the porous medium equation in . J. Evol. Equ. 2011, 11(2):429-455. 10.1007/s00028-010-0097-4MATHMathSciNetView ArticleGoogle Scholar
- Wang L, Yin J, Jin C: ω -Limit sets for porous medium equation with initial data in some weighted spaces. Discrete Contin. Dyn. Syst., Ser. B 2013, 18(1):223-236.MATHMathSciNetView ArticleGoogle Scholar
- DiBenedetto E: Degenerate Parabolic Equations. Springer, New York; 1993.MATHView ArticleGoogle 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/BF02801989MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.