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
In this paper, we investigate the large time behavior of non-negative solutions for the following initial-boundary value problem:
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.
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 , Andreucci proved the existence of solutions for the following degenerate parabolic equation with initial data measures:
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  studied the large time behavior of solutions to the following Cauchy problem:
and he gave the temporal decay estimates.
In , Stinner investigated the asymptotic behavior of solutions for the following one space dimensional viscous Hamilton-Jacobi equation:
In higher dimensional case, Barles et al. studied the large time behavior of solutions for the following initial-boundary value problem:
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.
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:
Remark 1.1 Compared to the results in , 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.
Next, we claim that
Therefore, we have
By the comparison principle , we have
According to (2.8), the parameter δ can be taken to be arbitrarily small in (2.9). Therefore, we deduce that
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).
satisfying (2.2) and (2.3). Then
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.
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)
By the comparison principle, we have
The proof of Lemma 3.1 is complete. □
Lemma 3.2 (Upper bound)
By the comparison principle, we have
The proof of Lemma 3.2 is complete. □
Lemma 3.3 (Hölder estimate)
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
Moreover, we assume that
By the comparison principle  we have
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.
Proofs of Theorems 1.1 and 1.2 The proofs are based on the ideas in , 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
According to the nonnegativity of , it follows from the comparison principle  that
Then v is a viscosity solution to the following problem:
In addition, owing to Lemmas 3.1-3.3 and (3.14), we have
and the half-relaxed limits
Case 1. . We use the stability of semicontinuous viscosity solutions  to deduce from (3.19) that
In addition, by , in as . Moreover, it follows from (3.16) and the definition of and that
Since in Ω by , 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).
Therefore, we infer from the scaling transformation that
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  to deduce from (3.19) that
In addition, by , in as . Moreover, it follows from (3.16) and the definition of and that
Since in Ω by  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
Thus, we infer from the scaling transformation that
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.
The proof of Lemma 4.1 is complete. □
Since on and for by (4.5), we infer from the comparison principle  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 . 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
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 .
By a direct calculation, we have
By Young’s inequality, we obtain
Thus, by (5.4) and (5.5), we obtain
Therefore, it follows from (5.6) and (5.7) that
as long as
The proof of Theorem 1.4 is complete. □
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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).
Andreucci, D: Degenerate parabolic equations with initial data measures. Trans. Am. Math. Soc.. 349, 3911–3923 (1997). Publisher Full Text
Gilding, BH, Guedda, M, Kersner, R: The Cauchy problem for . J. Math. Anal. Appl.. 284, 733–755 (2003). Publisher Full Text
Li, YX, Souplet, P: Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. Commun. Math. Phys.. 293, 499–517 (2010). Publisher Full Text
Crandall, MG, Ishii, H, Lions, PL: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.). 27, 1–67 (1992). Publisher Full Text
Ding, JT, Guo, BZ: Global existence and blow-up solutions for quasilinear reaction-diffusion equations with a gradient term. Appl. Math. Lett.. 24, 936–942 (2011). Publisher Full Text
Souplet, P, Zhang, QS: Global solutions of inhomogeneous Hamilton-Jacobi equations. J. Anal. Math.. 99, 355–396 (2006). Publisher Full Text
Xia, L, Yao, ZA: Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation. J. Math. Anal. Appl.. 358, 182–188 (2009). Publisher Full Text
Zhou, WS, Lei, PD: A one-dimensional nonlinear heat equation with a singular term. J. Math. Anal. Appl.. 368, 711–726 (2010). Publisher Full Text
Barles, G, Souganidis, PE: On the large time behaviour of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal.. 31, 925–939 (2000). Publisher Full Text
Benachour, S, Dabuleanu, S: Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition. J. Differ. Equ.. 216, 223–258 (2005). Publisher Full Text
Laurençot, P: Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions. Pac. J. Math.. 230, 347–364 (2007). Publisher Full Text
Namah, G, Roquejoffre, JM: Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations. Commun. Partial Differ. Equ.. 24, 883–893 (1999). Publisher Full Text
Qi, YW, Wang, MX: The self-similar profiles of generalized KPZ equation. Pac. J. Math.. 201, 223–240 (2001). Publisher Full Text
Roquejoffre, JM: Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. J. Math. Pures Appl.. 80, 85–104 (2001). Publisher Full Text
Shi, PH, Wang, MX: Global solution of the fast diffusion equation with gradient absorption terms. J. Math. Anal. Appl.. 326, 602–621 (2007). Publisher Full Text
Gilding, BH: The Cauchy problem for : large-time behaviour. J. Math. Pures Appl.. 84, 753–785 (2005). Publisher Full Text
Stinner, C: Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion. J. Differ. Equ.. 248, 209–228 (2010). Publisher Full Text
Laurençot, P, Stinner, C: Convergence to separate variable solutions for a degenerate parabolic equation with gradient source. J. Dyn. Differ. Equ.. 24, 29–49 (2012). Publisher Full Text
Mu, CL, Li, YH, Wang, Y: Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values. Nonlinear Anal., Real World Appl.. 11, 198–206 (2010). Publisher Full Text
Da Lio, F: Comparison results for quasilinear equations in annular domains and applications. Commun. Partial Differ. Equ.. 27, 283–323 (2002). Publisher Full Text
Guo, JS, Guo, YY: On a fast diffusion equation with source. Tohoku Math. J.. 53, 571–579 (2001). Publisher Full Text
Li, YH, Mu, CL: Life span and a new critical exponent for a degenerate parabolic equation. J. Differ. Equ.. 207, 392–406 (2004). Publisher Full Text
Mukai, K, Mochizuki, K, Huang, Q: Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values. Nonlinear Anal. TMA. 39, 33–45 (2000). Publisher Full Text