Complicated asymptotic behavior of solutions for a porous medium equation with nonlinear sources

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.

In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem of the porous medium equation with nonlinear sources

where and with .

It is well known that any positive solutions of problem (1.1)-(1.2) blow up in finite time if [1-3], while positive global solutions do exist if [4-7]. 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:

Moreover, if , then

uniformly on , where . Here is a semigroup generated by the Cauchy problem of the porous medium equation

and .

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 [10-14].

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.

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).

Definition 2.1[16,17]

By a weak solution of problem (1.1)-(1.2) in , we mean a function in such that

1. in and for each .

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).

Lemma 2.1[16,17]

Suppose that for, are supersolution and subsolution of the problem (1.1)-(1.2), respectively. If

then, for all,

To study the asymptotic behavior of solutions for problem (1.1)-(1.2), we adopt the space and as that in [16-18]. For any and , the is defined as

with the obvious norm and the is given by

with the norm . Here

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[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 ,

see details in [19,20].

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

Now letting to satisfy

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. □

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 ,

and for all ,

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 cut-off function defined on relatively to , and is selected large enough to satisfy

Notice first that if , then

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

uniformly on . So, we complete the proof of Theorem 3.1. □

The authors declare that they have no competing interests.

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.

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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