Large time behavior of solutions for the porous medium equation with a nonlinear gradient source
© Li et al.; licensee Springer. 2014
Received: 23 December 2013
Accepted: 3 April 2014
Published: 6 May 2014
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.
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 ).
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.
where , , , and .
and he gave the temporal decay estimates.
where , , and , and he proved that these solutions converge to the steady states by Lyapunov functional.
where , , and they showed that the non-negative radially symmetric solutions converge to the stationary solution.
Recently, in , 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.
where is defined in (1.12).
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.
For large enough, it is easy to see that is a non-empty open subset of .
It follows from and w vanishes on ∂ Ω by (2.2) that .
Letting , we conclude that zero is a cluster point of as . The claim (2.7) follows from the monotonicity of .
for and satisfying (2.8).
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 2.1 There is at most one positive viscosity solution to (1.9).
By the similar argument, we have the following result to (1.12).
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 3.1 Assume that , , , and satisfies (1.2). Then there exists a unique solution to (1.1) in the sense of Definition 1.1.
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)
The proof of Lemma 3.1 is complete. □
Lemma 3.2 (Upper bound)
The proof of Lemma 3.2 is complete. □
Lemma 3.3 (Hölder estimate)
for and .
where , , and are defined in Lemmas 3.1 and 3.2, respectively.
where , and and are defined in Lemmas 3.1 and 3.2, respectively.
The proof of Lemma 3.3 is complete. □
We next proceed as in  to deduce the Hölder continuity of from Lemma 3.3. Therefore, we obtain the following corollary.
Next, we shall give proofs of Theorems 1.1 and 1.2. To do this, we distinguish the two cases and .
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).
Finally, Corollary 3.1 gives the last statement of Theorem 1.1. The proof of Theorem 1.1 is complete.
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 . To do this, we need the following lemma.
where the parameter is defined in (1.10). Then is a solution to (1.1) with the initial data and such that for . Moreover, is a supersolution to (1.1) for .
Proof According to the definition of in (1.10), it is easy to see that for .
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 .
The proof of Lemma 4.1 is complete. □
where is 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).
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 . The proof of Proposition 4.1 is complete. □
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 .
Therefore, increases and remains above for all .
Hence, it follows from (5.10) and (5.11) that the solution of (1.1) blows up in finite time, .
The proof of Theorem 1.4 is complete. □
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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