Abstract
This paper deals with the large time behavior of nonnegative 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; blowup1 Introduction
In this paper, we investigate the large time behavior of nonnegative solutions for the following initialboundary value problem:
where , , Ω is a bounded domain of () with smooth boundary ∂Ω, and the initial function is
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 HamiltonJacobi 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 wellposedness of the initialboundary value problem (1.1) is the theory of viscosity solutions (see [46]), so we first define the notion of solutions.
Definition 1.1 A nonnegative function is called a solution of (1.1), if is a viscosity solution to (1.1) in and satisfies
Under some assumptions, the global (local) existence in time, uniqueness and regularity of solutions to reactiondiffusion equations with gradient terms have been extensively investigated by many authors (see [712] and the references therein). In particular, in [1], 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 nonnegative solutions to (1.4) with homogeneous Dirichlet boundary conditions. In recent years, many authors have investigated the asymptotic behavior of solutions to the viscous HamiltonJacobi equations (see [3,4,7,8,10,1321] and the references therein). For example, for the special case , Gilding [22] studied the large time behavior of solutions to the following Cauchy problem:
and he gave the temporal decay estimates.
In [23], Stinner investigated the asymptotic behavior of solutions for the following one space dimensional viscous HamiltonJacobi equation:
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 initialboundary value problem:
where , , and they showed that the nonnegative 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 selfsimilar transformation method, and the halfrelaxed limits technique used in [6,25,26], we investigate the asymptotic behavior of nonnegative 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
whereis the unique positive solution to
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
whereis the unique positive solution to
Theorem 1.3Let, , and. Assume thatsatisfies (1.2) and suppose further that there existssatisfying (1.2) such that
whereis defined in (1.10). Then there exists a unique solutionto (1.1) in the sense of Definition 1.1 such that
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 blowup time is given as follows:
Remark 1.1 Compared to the results in [25], we extend the results of pLaplacian 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 blowup time. But the lower estimate of the blowup 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 halfrelaxed 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 halfrelaxed 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 blowup 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 nonnegative 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 halfrelaxed 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
such that
and
Then
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 nonempty 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
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
It follows from and w vanishes on ∂Ω by (2.2) that .
Next, we claim that
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
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
According to (2.8), the parameter δ can be taken to be arbitrarily small in (2.9). Therefore, we deduce that
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
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 wellposedness and large time behavior of solutions to (1.1) for , and we prove Theorems 1.11.2. To do this, we first obtain the wellposedness 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 halfrelaxed limits technique is applied to show the expected convergence after introducing selfsimilar 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
Proof Assume that and such that . For , we define the function
where and satisfies the following condition:
Since , the function is smooth in . Moreover, according the condition , , and , we have and . Therefore, for and , it follows from (3.2) that
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
Proof Assume that and such that . For , we define the function
where the positive constants , R, and δ satisfy the following condition:
Since , the function is smooth in . Moreover, for and , it follows from (3.5) and that
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
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
Next, we define the open subset of by
where satisfies , and denote the function
Moreover, we assume that
and
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 3.13.2 and (3.9), we have
By the comparison principle [6] we have
Consequently,
Finally, we consider and . If , it follows from Lemmas 3.13.2 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
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
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
According to the nonnegativity of , it follows from the comparison principle [6] that
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:
In addition, owing to Lemmas 3.13.3 and (3.14), we have
and
and the halfrelaxed limits
By (3.16), it is easy to see that and are welldefined and do not depend on . Moreover, it readily follows from (3.15) and (3.17) that
By a direct computation, is a solution to the following initialboundary problem:
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
and
In addition, by [30], in as . Moreover, it follows from (3.16) and the definition of and that
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.,
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 [6] to deduce from (3.19) that
and
In addition, by [30], in as . Moreover, it follows from (3.16) and the definition of and that
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.,
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 wellposedness 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
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
Moreover, introducing for , the function has a local minimum at such that
i.e.,
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
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
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
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 blowup condition corresponding to (1.1), we have to modify the function used in [3133], 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 blowup time of the solution. For , we denote
By a direct calculation, we have
By Young’s inequality, we obtain
Since , it follows from (5.2), (5.3), and Poincaré’s inequality that
According to , , and Hölder’s inequality, we have
Thus, by (5.4) and (5.5), we obtain
Owing to , , , and Jensen’s inequality, we have
Therefore, it follows from (5.6) and (5.7) that
as long as
Taking
Since the initial data satisfies
we have
Therefore, increases and remains above for all .
By (5.8), integrating over yields
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 blowup time T of the solution as follows:
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).
References

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: Singlepoint gradient blowup on the boundary for diffusive HamiltonJacobi equations in planar domains. Commun. Math. Phys.. 293, 499–517 (2010). Publisher Full Text

Bardi, M, CapuzzoDolcetta, I: Optimal Control and Viscosity Solutions of HamiltonJacobiBellman Equations, Birkhäuser, Boston (1997)

Barles, G: Solutions de Viscosité des Equations d’HamiltonJacobi, Springer, Berlin (1994)

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

Arrieta, JM, Bernal, AR, Souplet, P: Boundedness of global solutions for nonlinear parabolic equations involving gradient blowup phenomena. Ann. Sc. Norm. Super. Pisa, Cl. Sci.. 3, 1–15 (2004)

Benachour, SD, DăbuleanuHapca, S, Laurençot, P: Decay estimates for a viscous HamiltonJacobi equation with homogeneous Dirichlet boundary conditions. Asymptot. Anal.. 51, 209–229 (2007)

Ding, JT, Guo, BZ: Global existence and blowup solutions for quasilinear reactiondiffusion equations with a gradient term. Appl. Math. Lett.. 24, 936–942 (2011). Publisher Full Text

Souplet, P, Zhang, QS: Global solutions of inhomogeneous HamiltonJacobi 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 onedimensional 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 HamiltonJacobi equations. SIAM J. Math. Anal.. 31, 925–939 (2000). Publisher Full Text

Benachour, S, Dabuleanu, S: Large time behavior for a viscous HamiltonJacobi equation with Neumann boundary condition. J. Differ. Equ.. 216, 223–258 (2005). Publisher Full Text

Laurençot, P: Convergence to steady states for a onedimensional viscous HamiltonJacobi 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 HamiltonJacobi equations. Commun. Partial Differ. Equ.. 24, 883–893 (1999). Publisher Full Text

Qi, YW, Wang, MX: The selfsimilar profiles of generalized KPZ equation. Pac. J. Math.. 201, 223–240 (2001). Publisher Full Text

Quittner, P, Souplet, P: Superlinear Parabolic Problems. Blowup, Global Existence and Steady States, Birkhäuser, Basel (2007)

Roquejoffre, JM: Convergence to steady states or periodic solutions in a class of HamiltonJacobi 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

Tchamba, TT: Large time behavior of solutions of viscous HamiltonJacobi equations with superquadratic Hamiltonian. Asymptot. Anal.. 66, 161–186 (2010)

Gilding, BH: The Cauchy problem for : largetime behaviour. J. Math. Pures Appl.. 84, 753–785 (2005). Publisher Full Text

Stinner, C: Convergence to steady states in a viscous HamiltonJacobi equation with degenerate diffusion. J. Differ. Equ.. 248, 209–228 (2010). Publisher Full Text

Barles, G, Laurençot, P, Stinner, C: Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive HamiltonJacobi equation. Asymptot. Anal.. 67, 229–250 (2010)

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

Laurençot, P, Stinner, C: Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions. Commun. Partial Differ. Equ.. 36, 532–546 (2011)

Da Lio, F: Comparison results for quasilinear equations in annular domains and applications. Commun. Partial Differ. Equ.. 27, 283–323 (2002). Publisher Full Text

Kawohl, B, Kutev, N: Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations. Funkc. Ekvacioj. 43, 241–253 (2000)

Manfredi, JJ, Vespri, V: Large time behavior of solutions to a class of doubly nonlinear parabolic equations. Electron. J. Differ. Equ.. 2, 1–17 (1994)

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