Research

# Large time behavior of solutions for the porous medium equation with a nonlinear gradient source

Nan Li1, Pan Zheng23*, Chunlai Mu2 and Iftikhar Ahmed2

Author Affiliations

1 Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, P.R. China

2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China

3 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P.R. China

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Boundary Value Problems 2014, 2014:98  doi:10.1186/1687-2770-2014-98

 Received: 23 December 2013 Accepted: 3 April 2014 Published: 6 May 2014

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 credited.

### Abstract

This paper deals with the large time behavior of non-negative solutions for the porous medium equation with a nonlinear gradient source , , where and . When , we prove that the global solution converges to the separate variable solution . While , we note that the global solution converges to the separate variable solution . Moreover, when , we show that the global solution also converges to the separate variable solution for the small initial data , and we find that the solution blows up in finite time for the large initial data .

MSC: 35K55, 35K65, 35B40.

##### Keywords:
large time behavior; separate variable solution; porous medium equation; gradient source; blow-up

### 1 Introduction

In this paper, we investigate the large time behavior of non-negative solutions for the following initial-boundary value problem:

(1.1)

where , , Ω is a bounded domain of () with smooth boundary Ω, and the initial function is

(1.2)

Equation (1.1) arises in the study of the growth of surfaces and it has been considered as a mathematical model for a variety of physical problems (see [1,2]). For instance, in ballistic deposition processes, the evolution of the profile of a growing interface is described by the diffusive Hamilton-Jacobi type equation (1.1) with (see [3]).

One of the particular feature of problem (1.1) is that the equation is a slow diffusion equation with nonlinear source term depending on the gradient of a power of the solution. In general, there is no classical solution. Therefore, it turns out that a suitable framework for the well-posedness of the initial-boundary value problem (1.1) is the theory of viscosity solutions (see [4-6]), so we first define the notion of solutions.

Definition 1.1 A non-negative function is called a solution of (1.1), if is a viscosity solution to (1.1) in and satisfies

(1.3)

Under some assumptions, the global (local) existence in time, uniqueness and regularity of solutions to reaction-diffusion equations with gradient terms have been extensively investigated by many authors (see [7-12] and the references therein). In particular, in [1], Andreucci proved the existence of solutions for the following degenerate parabolic equation with initial data measures:

(1.4)

where , , , and .

The main purpose of this paper is to further study the large time behavior of non-negative solutions to (1.4) with homogeneous Dirichlet boundary conditions. In recent years, many authors have investigated the asymptotic behavior of solutions to the viscous Hamilton-Jacobi equations (see [3,4,7,8,10,13-21] and the references therein). For example, for the special case , Gilding [22] studied the large time behavior of solutions to the following Cauchy problem:

(1.5)

and he gave the temporal decay estimates.

In [23], Stinner investigated the asymptotic behavior of solutions for the following one space dimensional viscous Hamilton-Jacobi equation:

(1.6)

where , , and , and he proved that these solutions converge to the steady states by Lyapunov functional.

In higher dimensional case, Barles et al.[24] studied the large time behavior of solutions for the following initial-boundary value problem:

(1.7)

where , , and they showed that the non-negative radially symmetric solutions converge to the stationary solution.

Recently, in [25], Laurencot et al. extended the case of the problem (1.7) to the case , and derived that these solutions converge to two different separate variables solutions according to the cases and in the general bounded domain , respectively.

Motivated by the above work, by using the modified comparison argument, the self-similar transformation method, and the half-relaxed limits technique used in [6,25,26], we investigate the asymptotic behavior of non-negative solutions to (1.1). Our main results in this paper are stated as follows.

Theorem 1.1Let, , and. Assume thatsatisfies (1.2). Then there exists a unique solutionto (1.1) in the sense of Definition 1.1 such that

(1.8)

whereis the unique positive solution to

(1.9)

Moreover, we havefor alland

(1.10)

Theorem 1.2Let, , and. Assume thatsatisfies (1.2). Then there exists a unique solutionto (1.1) in the sense of Definition 1.1 such that

(1.11)

whereis the unique positive solution to

(1.12)

Theorem 1.3Let, , and. Assume thatsatisfies (1.2) and suppose further that there existssatisfying (1.2) such that

(1.13)

whereis defined in (1.10). Then there exists a unique solutionto (1.1) in the sense of Definition 1.1 such that

(1.14)

whereis defined in (1.12).

Theorem 1.4Let, , and. Assume that there exists a positive constantdepending only onm, l, q, s, andεsuch that, where, , andwith. Then the solutionof the problem (1.1) blows up in finite time in the sense of weak solution. Moreover, the upper bound of blow-up time is given as follows:

(1.15)

Remark 1.1 Compared to the results in [25], we extend the results of p-Laplacian equation to the porous medium equation (1.1) with a nonlinear gradient source.

Remark 1.2 In Theorem 1.4, we only give an upper estimate of the blow-up time. But the lower estimate of the blow-up time is an open problem.

This paper is organized as follows. In Section 2, we establish the comparison lemmas to prove the uniqueness of the positive solution to (1.9) and the identification of the half-relaxed limits. In Section 3, using the comparison principle, we construct the global solutions to obtain the upper bound and Hölder estimate of solutions to (1.1), and we prove Theorems 1.1 and 1.2 by the half-relaxed limits method. Moreover, we give the large time behavior of solutions to (1.1) with the small initial data for , and we prove Theorem 1.3 in Section 4. Finally, we obtain the blow-up case, and we prove Theorem 1.4 in Section 5.

### 2 Comparison lemmas

In this section, we establish the following comparison lemma between positive supersolutions and non-negative subsolutions to the elliptic equation in (1.9): which is an important tool for the uniqueness of the positive solution to (1.9) and the identification of the half-relaxed limits later.

Lemma 2.1Let, , and. Assume thatandare respectively a bounded upper semicontinuous (usc) viscosity subsolution and a bounded lower semicontinuous (lsc) viscosity supersolution to

(2.1)

such that

(2.2)

and

(2.3)

Then

(2.4)

Proof The proof is based on the idea as in [25,27], but with different auxiliary functions.

For large enough, it is easy to see that is a non-empty open subset of .

Since is compact and W is lower semicontinuous, the function W has a minimum in . By the positivity (2.3) of W in , we have

(2.5)

Similarly, the compactness of , the upper semicontinuity and boundedness of w imply that w has at least one point of maximum in and we set

(2.6)

It follows from and w vanishes on Ω by (2.2) that .

Next, we claim that

(2.7)

Indeed, owing to the compactness of and the definition of , there exist and a subsequence of (not relabeled) such that as . Since , we deduce from the upper semicontinuity of w that

Given small enough, there exists large enough such that

Therefore, we have

Thus

Letting , we conclude that zero is a cluster point of as . The claim (2.7) follows from the monotonicity of .

Now, fix . For and , we define

and

It follows from the assumptions on w and W that and are, respectively, a bounded usc viscosity subsolution and a bounded lsc viscosity supersolution to

and satisfy

Moreover, if

(2.8)

then it follows from (2.5) and (2.8) that, for ,

For , we deduce from (2.6) that

By the comparison principle [6], we have

(2.9)

for and satisfying (2.8).

According to (2.8), the parameter δ can be taken to be arbitrarily small in (2.9). Therefore, we deduce that

for .

Passing to the limit as , it follows from (2.7) that

Finally, let and take ; then we obtain

The proof of Lemma 2.1 is complete. □

A straightforward consequence of Lemma 2.1 is the uniqueness of the solution to (1.9).

Corollary 2.1There is at most one positive viscosity solution to (1.9).

By the similar argument, we have the following result to (1.12).

Lemma 2.2Letandbe respectively a bounded upper semicontinuous (usc) viscosity subsolution and a bounded lower semicontinuous (lsc) viscosity supersolution to

(2.10)

satisfying (2.2) and (2.3). Then

(2.11)

Proof The proof is similar as in Lemma 2.1, so we omit it here. □

### 3 Proofs of Theorems 1.1 and 1.2

In this section, we obtain the well-posedness and large time behavior of solutions to (1.1) for , and we prove Theorems 1.1-1.2. To do this, we first obtain the well-posedness to (1.1) by the following proposition.

Proposition 3.1Assume that, , , andsatisfies (1.2). Then there exists a unique solutionto (1.1) in the sense of Definition 1.1.

Proof The idea of the proof is same as in [25,28], so we omit here. □

Next, in order to prove the large time behavior of the solution to (1.1), we shall need several steps: Step 1, we will find that the temporal decay rate of is indeed . Step 2, we prove the boundary estimates for the large time which guarantee that no loss of boundary condition occurs throughout the time evolution. Step 3, the half-relaxed limits technique is applied to show the expected convergence after introducing self-similar variables. The approach is developed by Laurençot and Stinner in [25,27]. To do this, we need the following lemmas.

Lemma 3.1 (Upper bound)

Assume that, , , and the initial datasatisfies (1.2). Then there existsdepending only onm, l, q, Ω, andsuch that

(3.1)

Proof Assume that and such that . For , we define the function

where and satisfies the following condition:

(3.2)

Since , the function is -smooth in . Moreover, according the condition , , and , we have and . Therefore, for and , it follows from (3.2) that

(3.3)

Hence, the condition (3.2) guarantees that is a supersolution to (1.1) in . In addition, since for , for , we deduce from (3.2) that

and

By the comparison principle, we have

The proof of Lemma 3.1 is complete. □

Lemma 3.2 (Upper bound)

Assume that, , , and the initial datasatisfies (1.2). Then there existsdepending only onm, l, q, Ω, andsuch that

(3.4)

Proof Assume that and such that . For , we define the function

where the positive constants , R, and δ satisfy the following condition:

(3.5)

Since , the function is -smooth in . Moreover, for and , it follows from (3.5) and that

(3.6)

Therefore, the condition (3.5) guarantees that is a supersolution to (1.1) in . In addition, since for , for , we deduce from (3.5) that

and

By the comparison principle, we have

The proof of Lemma 3.2 is complete. □

Lemma 3.3 (Hölder estimate)

Assume that, , , and the initial datasatisfies (1.2). Then there existsdepending only onm, l, q, Ω, andsuch that

(3.7)

Proof Since the boundary Ω of Ω is smooth, there exists such that for each , there exists satisfying and . It follows from the initial data condition that is Lipschitz continuous, i.e., there exists such that

(3.8)

Next, we define the open subset of by

where satisfies , and denote the function

for and .

Moreover, we assume that

(3.9)

and

(3.10)

where , , and are defined in Lemmas 3.1 and 3.2, respectively.

Since , the function is -smooth in . For , we have . By a direct computation, we infer from (3.9)-(3.10) that

Therefore, is a supersolution to (1.1) in . In addition, it follows from (3.8)-(3.10) and the mean value theorem that

Moreover, for , we have either or . If , then we have

If , by Lemmas 3.1-3.2 and (3.9), we have

By the comparison principle [6] we have

Consequently,

for .

Since , we have

(3.11)

Finally, we consider and . If , it follows from Lemmas 3.1-3.2 that

where , and and are defined in Lemmas 3.1 and 3.2, respectively.

If , let satisfy and . Since , we have

which implies . Therefore, we deduce from (3.11) that

Choosing , we have

The proof of Lemma 3.3 is complete. □

We next proceed as in [29] to deduce the Hölder continuity of from Lemma 3.3. Therefore, we obtain the following corollary.

Corollary 3.1Assume that, , , and the initial datasatisfies (1.2). Then there existsdepending only onm, l, q, Ω, andsuch that

(3.12)

Proof The proof is similar to the argument in [25,29], so we omit here. □

Proofs of Theorems 1.1 and 1.2 The proofs are based on the ideas in [25], but we give the details of the argument for the reader’s convenience. Let be the solution to the porous medium equation with homogeneous Dirichlet boundary conditions

(3.13)

According to the nonnegativity of , it follows from the comparison principle [6] that

(3.14)

We introduce the scaling variable for and denote the new unknown function v and V by

and

Then v is a viscosity solution to the following problem:

(3.15)

In addition, owing to Lemmas 3.1-3.3 and (3.14), we have

(3.16)

and

(3.17)

Next, for , we define

and the half-relaxed limits

for .

By (3.16), it is easy to see that and are well-defined and do not depend on . Moreover, it readily follows from (3.15) and (3.17) that

(3.18)

By a direct computation, is a solution to the following initial-boundary problem:

(3.19)

Next, we shall give proofs of Theorems 1.1 and 1.2. To do this, we distinguish the two cases and .

Case 1. . We use the stability of semicontinuous viscosity solutions [6] to deduce from (3.19) that

(3.20)

and

(3.21)

In addition, by [30], in as . Moreover, it follows from (3.16) and the definition of and that

(3.22)

Since in Ω by [30], we deduce from (3.22) that and are positive and bounded in Ω and vanish on Ω by (3.18). Owing to (3.20) and (3.21), we infer from Lemma 2.1 that

By (3.22), we have

Setting , we deduce from (3.18), (3.20), (3.21), and (3.22) that is a positive viscosity solution to (2.1) and solves (1.9). Therefore, the existence of a positive solution to (1.9) is proved. Moreover, by Corollary 2.1, we obtain the uniqueness of solution to (1.9).

Furthermore, it follows from the equality that

i.e.,

(3.23)

Therefore, we infer from the scaling transformation that

(3.24)

Finally, Corollary 3.1 gives the last statement of Theorem 1.1. The proof of Theorem 1.1 is complete.

Case 2. . We use once more the stability of semicontinuous viscosity solutions [6] to deduce from (3.19) that

(3.25)

and

(3.26)

In addition, by [30], in as . Moreover, it follows from (3.16) and the definition of and that

(3.27)

Since in Ω by [30] and is a solution to (2.10), we can apply Lemma 2.2 to conclude that in . Owing to (3.27), we have

Therefore, we deduce from (3.19) that

i.e.,

(3.28)

Thus, we infer from the scaling transformation that

(3.29)

The proof of Theorem 1.2 is complete. □

### 4 Proof of Theorem 1.3

In this section, we shall consider the well-posedness and the large time behavior of solutions to (1.1) with the small initial data for by the method used in [25]. To do this, we need the following lemma.

Lemma 4.1Letand. Assume that G is the corresponding solution to (1.1) with the initial datasatisfying (1.2) for the case, and denote

(4.1)

where the parameteris defined in (1.10). Thenis a solution to (1.1) with the initial dataandsuch thatfor. Moreover, is a supersolution to (1.1) for.

Proof According to the definition of in (1.10), it is easy to see that for .

Next, let and has a local minimum at . Since ℱ is smooth with respect to the time variable and Hölder continuous with respect to the space variable, we obtain

(4.2)

Moreover, introducing for , the function has a local minimum at such that

i.e.,

(4.3)

Therefore, is a supersolution to (1.1) with . In a similar way, it can be shown that is also a subsolution. Hence, is a solution to (1.1) with .

Furthermore, we deduce from (4.2), (4.3), and that

(4.4)

The proof of Lemma 4.1 is complete. □

Proposition 4.1Let, , and. Assume that the initial datasatisfies (1.2), moreover, there existssatisfying (1.2) such that

(4.5)

where the parameteris defined in (1.10). Then there exists a unique solution u to (1.1) in the sense of Definition 1.1 and it satisfies

(4.6)

whereis defined in (4.1).

Proof On the one hand, the solution U to the porous medium equation (3.13) is clearly a subsolution to (1.1) in .

On the other hand, it follows from Lemma 4.1 that the function is a supersolution to (1.1) in . Therefore, is a supersolution to (3.13).

Since on and for by (4.5), we infer from the comparison principle [6] that

This property and the simultaneous vanishing of U and ℱ on allow us to use the classical Perron method to establish the existence of a solution to (1.1) in the sense of Definition 1.1 which satisfies (4.6). The uniqueness next follows from the comparison principle [6]. The proof of Proposition 4.1 is complete. □

Proof of Theorem 1.3 We notice that Lemma 3.2 is also valid in that case. It readily follows from Lemma 3.3 and Proposition 4.1 that

The convergence proof is similar to that performed in the proof of Theorem 1.2 for . The proof of Theorem 1.3 is complete. □

### 5 Proof of Theorem 1.4

In this section, when , we shall prove that the solution of (1.1) blows up in finite time for the large initial data in the sense of weak solution by the method used in [26].

In order to obtain a blow-up condition corresponding to (1.1), we have to modify the function used in [31-33], and introduce a test function as follows:

Proof of Theorem 1.4 Suppose that is the solution of the problem (1.1) and T is the blow-up time of the solution. For , we denote

(5.1)

By a direct calculation, we have

(5.2)

By Young’s inequality, we obtain

(5.3)

Since , it follows from (5.2), (5.3), and Poincaré’s inequality that

(5.4)

According to , , and Hölder’s inequality, we have

(5.5)

Thus, by (5.4) and (5.5), we obtain

(5.6)

Owing to , , , and Jensen’s inequality, we have

(5.7)

Therefore, it follows from (5.6) and (5.7) that

(5.8)

as long as

(5.9)

Taking

Since the initial data satisfies

we have

(5.10)

Therefore, increases and remains above for all .

By (5.8), integrating over yields

(5.11)

Hence, it follows from (5.10) and (5.11) that the solution of (1.1) blows up in finite time, .

Moreover, by (5.10), we obtain the upper estimate on the blow-up time T of the solution as follows:

(5.12)

The proof of Theorem 1.4 is complete. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to express sincere gratitude to the referees for their valuable suggestions and comments on the original manuscript. The first author is supported by Scientific Research Fund of Sichuan Provincial Science and Technology Department (2014JY0098); the second author is supported in part by the Fundamental Research Funds for the Central Universities, Project No. CDJXS 12 10 00 14; the third author is supported in part by NSF of China (11371384).

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