Abstract
Keywords:
complexity; asymptotic behavior; porous medium equation1 Introduction
In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem of the porous medium equation with nonlinear sources
It is well known that any positive solutions of problem (1.1)(1.2) blow up in finite time if [13], while positive global solutions do exist if [47]. In 2000, Mukai, Mochizuki and Huang in [6] found that if and and satisfies , then there exists a constant such that for , the solutions of problem (1.1)(1.2) with the initial value are global and the following estimate holds:
uniformly on , where . Here is a semigroup generated by the Cauchy problem of the porous medium equation
On the other hand, regarding problem (1.4)(1.5), in 2002, Vázquez and Zuazua [8] found that for any bounded sequence in , there exists an initial value and a sequence as such that uniformly on any compact subsets of . In our previous papers [9], for any bounded sequence in , we have shown that there exists an initial value and a sequence as such that
uniformly on , where and . For more details on the study of complicated asymptotic behavior of solutions for the heat equation and other evolution equations, we refer the readers to [1014].
In this paper, we are quite interested in the above mentioned same topic for the equation with strongly nonlinear sources, namely equation (1.1) with . We will show that for any , there is a constant and an initial value with such that for any , there exists a sequence as satisfying
uniformly on . Here . For this purpose, we first show that if the initial value , then the solutions are global and satisfy
One can easily see that (1.6) captures (1.3). From this, we can follow the framework by Kamin and Peletier [15] to prove that
So, we can get our results by following the framework in [9] 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).
By a weak solution of problem (1.1)(1.2) in , we mean a function in such that
2. For and any nonnegative which vanishes for large , the following equation holds:
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).
Suppose that for, are supersolution and subsolution of the problem (1.1)(1.2), respectively. If
To study the asymptotic behavior of solutions for problem (1.1)(1.2), we adopt the space and as that in [1618]. For any and , the is defined as
with the obvious norm and the is given by
Hence they are both Banach spaces. The existence and uniqueness of a weak solution of problem (1.4)(1.5) with the initialvalue is shown in [16,17], and this solution satisfies the following proposition.
Proposition 2.1[17]
Problem (1.4)(1.5) generates a continuous bounded semigroup ingiven by
In other words, . Moreover, if, then the semigroupis a contraction.
We now introduce the definitions of scalings and the commutative relations between the semigroup operators and the dilation operators. For any and , the dilation is defined as follows:
From the definitions of the dilation operator and the semigroup operator, we can get that for and ,
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.
Theorem 2.1Letand. There exists a constantsuch that for any, and, the solutionsof problem (1.1)(1.2) with the initial valueare global. Moreover, the following estimate holds:
whereis a constant dependent only onMandη.
Remark 2.1 Notice that if and , then . So, our results capture Theorem 3 in [6]. Here we use some ideas of them.
Proof To prove this theorem, we need the fact that if , then
which has been given in Lemma 2.6 of [20]. We give the proof here for completeness. In fact,
This means that . Therefore, from Proposition 2.1, we obtain that is well defined. Taking and in (2.2), we have
Now taking , and in (2.5), we obtain that
The fact that clearly means that
see [21]. This implies that for , the following limit holds:
Let
So,
Therefore,
as . This means that there exists an such that if , then
From (2.7), for , there exists a constant C such that
Combining (2.8) and (2.9), we have
By (2.6), we thus obtain that
So, we complete the proof of (2.4). Now taking
we get that
Therefore, by the comparison principle and (2.4), for all , we have
Since (see [17,21]), there exists a such that for all and ,
Combining this with (2.6) and using the comparison principle, we can get
In other words,
If , (2.3) clearly holds. In the rest of proof, we can assume that . The hypothesis indicates
Let
where is the constant given by (2.11). For , taking
and
we obtain from (2.11) that is an increasing function satisfying
and then taking
one can see that is a supersolution of the following problem:
Therefore,
(2.12) and (2.13) clearly mean that
From this and (2.14), we can get (2.3). So, we complete the proof of this theorem. □
3 Complicated asymptotic behavior
For any , let be as given by Theorem 2.1. We introduce
and
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.
Theorem 3.1Letand. Then there is a functionsuch that for any, there exists a sequenceassuch that
uniformly on. Hereis the solution of problem (1.1)(1.2).
To get this theorem, we need to prove the following lemma first.
Lemma 3.1Supposeand. Letube a solution of problem (1.1)(1.2). Ifwith, then
Proof
We first define the functions
and
where and . Using the comparison principle, we know that for ,
The results of Theorem 2.1 imply that
Here we have used the fact . So,
Now we estimate the integral
with in several steps. For any , we take λ large enough to satisfy and assume, without loss of generality, that in the rest of this proof. Then using the same method as above, we have
where . Similarly, we can get the integral estimates for , which have been given in [22]. By using the same methods as in [15], we can get that for
uniformly on any compact subset of . For any , we can obtain from (3.1) that there exists a constant satisfying
and
where and . Taking R as given by (3.4), from (3.3), there exists such that for all ,
Therefore, from (3.4)(3.6), we have
Now letting and in (3.7), we get that
So, we complete the proof of this lemma. □
Now we can prove our main result.
Proof of Theorem 3.1
Let
and
From the definition of , we obtain that there exists a countable set F such that
and for any and , there exists a function satisfying
Therefore, there exists a sequence such that
I. For any , there exists a subsequence of the sequence satisfying
II. There exists a constant satisfying
Now we can follow the methods given in [9] to construct an initial value as follows. Let
Here
is the cutoff function defined on relatively to , and is selected large enough to satisfy
and
By (3.9) and (3.10), we have
So, we have
Using the same method as that in [9], we can get that for any , there exists a sequence as such that
uniformly on . For any , from (1.2), we know that there exists a sequence such that
Therefore,
uniformly on any compact subset of . This uses the fact that the map is regularizing since the images of bounded sets are relatively compact subsets of for some in compact sets of [21]. And notice that . We thus obtain from Theorem 2.1 that for any , there exists such that if , then
and
Combining (3.12), (3.13) with (3.14), we thus have that
uniformly on . By Lemma 3.1, (3.11) and (3.15), we can get that for any , there exists a sequence as such that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The paper is the result of joint work of both authors who contributed equally to the final version of the paper. Both authors read and approved the final manuscript.
Acknowledgements
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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