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.
Keywordscomplexity asymptotic behavior porous medium equation
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).
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