Volume 2013, Issue 1 747613

Research Article

Open Access

Extinction and Nonextinction for the Fast Diffusion Equation

Chunlai Mu,

Chunlai Mu

College of Mathematics and Physics, Chongqing University, Chongqing 400044, China cqu.edu.cn

Search for more papers by this author

Li Yan,

Corresponding Author

Li Yan

[email protected]

College of Mathematics and Physics, Chongqing University, Chongqing 400044, China cqu.edu.cn

College of Elementary Education, Chongqing Normal University, Chongqing 400047, China cqnu.edu.cn

Search for more papers by this author

Yi-bin Xiao,

Corresponding Author

Yi-bin Xiao

[email protected]

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China uestc.edu.cn

Search for more papers by this author

Chunlai Mu,

Chunlai Mu

College of Mathematics and Physics, Chongqing University, Chongqing 400044, China cqu.edu.cn

Search for more papers by this author

Li Yan,

Corresponding Author

Li Yan

[email protected]

College of Mathematics and Physics, Chongqing University, Chongqing 400044, China cqu.edu.cn

College of Elementary Education, Chongqing Normal University, Chongqing 400047, China cqnu.edu.cn

Search for more papers by this author

Yi-bin Xiao,

Corresponding Author

Yi-bin Xiao

[email protected]

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China uestc.edu.cn

Search for more papers by this author

First published: 20 May 2013

https://doi.org/10.1155/2013/747613

Citations: 4

Academic Editor: Ru Dong Chen

Share a link

Email
Wechat
Bluesky

Abstract

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain of R^N with N > 2. For 0 < m < 1, under appropriate hypotheses, we show that m = p is the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of −Δ with homogeneous Dirichlet boundary.

1. Introduction

In this paper, we deal with the following fast diffusion equation with gradient absorption terms:

()

where 0 < m < 1, p > 0, λ > 0, Ω ⊂ R^N with N > 2 is an open bounded domain with smooth boundary, and

is a nonzero positive function.

Equation (1) appears in a lot of applications to describe the evolution of diffusion processes, in particular, fast diffusion for 0 < m < 1. In combustion theory, for instance, the function u(x, t) represents the temperature, the term Δu^m represents the thermal diffusion, and λ | ∇u|^p is a source.

Extinction and nonextinction are important properties for solutions of many evolutionary equations, especially for fast diffusion equations. In 1974, Kalashnikov [1] considered the Cauchy problem of equation u_t = Δu − u^p and firstly introduced the definition of extinction for its solution; that is, there exists a finite time T > 0 such that the solution is nontrivial for 0 < t < T, but u(x, t) ≡ 0 for all (x, t) ∈ Ω × (T, ∞). In this case, T is called the extinction time.

Since then, many authors became interested in the extinction and nonextinction of all kinds of evolutionary equations. For the following homogeneous Dirichlet boundary value problem:

()

Gu [2] obtained that the nontrivial solutions of the problem (2) vanish identically in a finite time if and only if 0 < q < 1, which implies that strong absorption will cause extinction in a finite time. More results on the extinction for the problem (2) have also been obtained by many researchers, and we can refer to [3–7] and the references therein. Because of the occurrence of such a phenomenon for a diffusion equation with a different absorption term, that is, the absorption term is a nonnegative function of ∇u instead of being a nonnegative function of u, Benachour et al. [8, 9] considered the following Cauchy problem for the viscous Hamilton-Jacobi equation:

()

They proved that the nonnegative classical solutions to the problem (3) are extinct in finite time and have noncompact support if 0 < q < 1.

Generally, in the problems (2) and (3), there is a comparison between the diffusion term and the absorption term, and the absorption is sufficiently strong to lead any bounded nonnegative solution to zero in finite time. However, while both the source −u^q in problem (2) and the source −|∇u|^q in problem (3) are called the “cool source,” the nonlinear source +|∇u|^q in problem (1) is physically called the “hot source.” As far as we know, the type of “hot source” has complicated influences on the properties of solutions compared with the case of “cool source” [10]. Thus, few works are concerned with extinction property for solutions to the evolutionary equations with “hot source.” In 2005, Li and Wu [10] gave some necessary and sufficient conditions of extinction for the solutions to the following problem with “hot source”:

()

where 0 < m < 1. They proved that if p > m, the solutions to the problem (4) with small initial data vanished in finite time and if p < m, the maximal solution to the problem (4) is positive for all t > 0. Recently, Tian and Mu [11] studied the following p-Laplacian equation with nonlinear “hot source”:

()

They obtained that for 1 < p < 2, q = p − 1 is the critical exponent of extinction for the weak solution to the problem (5). Also, they show that for 1 < p < 2 and q = p − 1, the extinction and nonextinction of the solution to the problem (5) depend strongly on the first eigenvalue of the problem −div (|∇u|^p−2∇u) = λ | u|^p−2u in Ω, u|_∂Ω = 0. For more results on the extinction properties of equations with “hot source,” we can refer to [12, 13] and the references therein.

By replacing the diffusion term u^p with the gradient absorption terms |∇u|^p in (4), in this paper, we devote to establish the conditions for the extinction of solution to the problem (1), which involves the “hot source” rather than the “cool source.”

2. Preliminaries

In this section, we will give some definitions and lemmas which is useful to the proof of our results in the next section. For our convenience, we first define some sets as follows:

()

It is well known that the problem (1) has no classical solution in general. We need to consider its weak solutions which is defined as follows.

Definition 1. For any T ≥ 0, a function u(x, t) ∈ E is called a weak solution to the problem (1) if the following equalities hold for any 0 < t₁ < t₂ < T and 0 ≤ φ ∈ E₀:

()

Remark 2. Similarly, to define a subsolution (resp., supersolution) (resp., ) of the problem (1), we need only to set (resp., ) in Ω, (resp., ) on ∂Ω × (0, ∞), and the first equality in (7) is replaced by ≤ (resp., ≥) for every φ(x) > 0.

As a useful tool in the proof of our main results in the next section, a comparison principle of the following problem is needed:

()

Under the following four hypotheses:

(H1)
φ, , , and , s.t. φ(x, t, 0) = 0 and φ_u(x, t, u) > 0 for all and u ≠ 0;
(H2)
and with f(x, t, 0) = 0 for all ;
(H3)
u₀ ∈ L^∞(Ω) with u₀ > 0;
(H4)
for any constant a > 0, if u ≥ 0, v ≥ a in and u, v ∈ L^∞(Ω_T), then the functions Φ and F defined almost everywhere in Ω_T by
()
belong to L^∞(Ω_T) and , respectively,

we have the following lemma on the comparison principle of the problem (8).

Lemma 3 (see [14].)Assume that the hypotheses (H1)–(H4) hold. Let u(x, t; u₀) and v(x, t; v₀) be nonnegative solutions of (8) on Ω_T with u₀ ≤ v₀. Assume further that for any t₁ < T, there is a constant a(t₁) > 0 such that v(x, t; v₀) ≥ a(t₁) for all . Then, u ≤ v almost everywhere on Ω_T.

3. Main Results

In this section, by applying the energy method introduced in [11] and the comparison result in Lemma 3 of the problem (8), we give the main results on the extinction and nonextinction of the solution to the problem (1).

3.1. Extinction of the Solution

In this subsection, we consider the extinction of the solution to the problem (1) and give the conditions for the extinction of the solution to the problem (1).

Theorem 4. Assume that 0 < m < 1, and let u(x, t) be a weak solution of the problem (1). If m < p ≤ 2/(3 − m), then, for sufficiently small initial data, there exists a finite time T, such that

()

for all (x, t) ∈ Ω × (T, +∞).

Proof. First of all, multiplying the first equation of the problem (1) by u^s−1 (s > 1) and integrating over Ω, we can obtain the following crucial equation to our proof:

()

Then, we prove the theorem by the following two cases.

First, we consider the case of (N − 2)/(N + 2) ≤ m < 1. Let s = 1 + m in (11), and we can get by the Hölder inequality and Poincare inequality that

()

where C₀ is the Sobolev embedding constant depending only on p and N.

Since p ≤ 2/(3 − m), 0 < m < 1, it follows from the Young’s inequality for any ε > 0 that

()

Also, since the assumptions p ≤ 2/(3 − m) and 0 < m < 1 imply that 2(p − pm + m)/(2 − p) ≤ 1 + m, we can obtain by the Hölder inequality that

()

Therefore, from the previous inequalities (11)–(14), we can obtain the following differential inequality:

()

Choose ε sufficiently small such that 1 − λm^−pε > 0 and initial data ∥u₀∥ sufficiently small such that

()

Thus, we can obtain that

()

where

By integrating the inequality (17), we can obtain that there exists a finite time such that

()

which implies that u(x, t) vanishes in finite time T.

Next, we consider the second case of 0 < m < (N − 2)/(N + 2) < (N − 2)/N. Let s = (N/2)(1 − m) > 1 in (11), and we can get by the Poincare inequality that

()

where C₃ is the Sobolev embedding constant depending only on N, m, and s. Also, by the similar calculations as the proof in the first case, we can get from the Young’s inequality for any η > 0 that

()

Choose η sufficiently small such that 4m(s − 1)/(m + s − 1) ² − λ(((1 + m)((N/2) − 1))/2) ^−pη > 0 and initial data ∥u₀∥ sufficiently small such that

()

Then, it follows from the inequalities (11) and (19)–(21) that

()

where

−

By integrating the inequality (22), we can obtain that there exists a finite time such that

()

which implies that u(x, t) vanishes in finite time T. This completes the proof of Theorem 4.

Theorem 5. If m = p ≤ 2/(3 − m), the solution to the problem (1) vanishes in finite time for λ sufficiently small.

Proof. When p = m, we note that the left sides of the inequalities (16) and (21) always equal 1. Thus, by a similar argument in the proof of Theorem 4, we can choose λ sufficiently small such that the inequalities (16) and (21) hold, which imply that there exist T < +∞ such that u(x, t) vanishes identically for all (x, t) ∈ Ω_T. This completes the proof of Theorem 5.

3.2. Nonextinction of the Solution

In this subsection, we investigate the conditions under which the solution u(x, t) of the problem (1) cannot become extinct.

Theorem 6. If p < m, the weak solution u(x, t) of (1) cannot vanish in finite time for any nonnegative initial date u₀ with λ being sufficiently large.

Proof. In order to prove Theorem 6, we first define two useful functions as follows. The first function satisfying max_x∈Ωϕ(x) = 1 is the associated eigenfunction with the principal eigenvalue λ₁ of the following problem:

()

For p < m, we define the second useful function as follows:

()

where a < b and C ∈ (0, (m − p)(a^1−p/b^1−m) ^1/(1−p)). It follows from [11] that the function g(t) satisfies the following properties:

()

Now, let v(x, t) = g(t)ϕ(x) ^1/m be a function Ω × (0, ∞). Next, we will show that v(x, t) is a subsolution of the problem (1). In fact, from the definitions of functions g(t), v(x, t) and the properties (26) of the function g(t), we have that

()

In order to prove that L(v(x, t)) < 0 which implies that v(x, t) is a subsolution of the problem (1), we only show that

()

Since 0 < g(t) < 1 and m > p, we have that

()

By choosing

, we can get that

()

which together with (29) implies that (28) holds. Therefore, v(x, t) is a subsolution of the problem (1). Moreover, since v(x, 0) = g(0)ϕ(x) = 0 ≤ u₀ in Ω and

, we can obtain by the comparison principle that u(x, t) ≥ v(x, t) > 0 in Ω × (0, +∞), which implies that the weak solution u(x, t) of (1) cannot vanish in finite time. This completes the proof of Theorem 6.

Remark 7. From Theorems 4–6, we observe that q = m is the critical exponent of extinction for the solution to the problem (1).

Acknowledgments

This work is supported in part by NSF of China (11101069, 11071266) and in part by the Natural Science Foundation Project of CQ CSTC (2010BB9218).

References

1 Kalashnikov A. S., The nature of the propagation of perturbations in problems of nonlinear heat conduction with absorption, USSR Computational Mathematics and Mathematical Physics. (1974) 14, no. 4, 70–85, MR0352702.
10.1016/0041-5553(74)90073-1
Google Scholar
2 Gu Y. G., Necessary and sufficient conditions for extinction of solutions to parabolic equations, Acta Mathematica Sinica. (1994) 37, no. 1, 73–79, MR1272507.
Google Scholar
3 Dibenedetto E., Degenerate Parabolic Equations, 1993, Springer, New York, NY, USA, https://doi.org/10.1007/978-1-4612-0895-2, MR1230384.
10.1007/978-1-4612-0895-2
Google Scholar
4 Evans L. C. and Knerr B. F., Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois Journal of Mathematics. (1979) 23, no. 1, 153–166, MR516577, ZBL0403.35052.
10.1215/ijm/1256048324
Google Scholar
5 Friedman A. and Herrero M. A., Extinction properties of semilinear heat equations with strong absorption, Journal of Mathematical Analysis and Applications. (1987) 124, no. 2, 530–546, https://doi.org/10.1016/0022-247X(87)90013-8, MR887008, ZBL0655.35050.
10.1016/0022-247X(87)90013-8
Web of Science® Google Scholar
6 Galaktionov V. A. and Vazquez J. L., Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Communications on Pure and Applied Mathematics. (1997) 50, no. 1, 1–67, MR1423231, ZBL0874.35057.
10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H
Web of Science® Google Scholar
7 Herrero M. A. and Velázquez J. J. L., Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption, Journal of Mathematical Analysis and Applications. (1992) 170, no. 2, 353–381, https://doi.org/10.1016/0022-247X(92)90024-8, MR1188559, ZBL0799.35111.
10.1016/0022-247X(92)90024-8
Web of Science® Google Scholar
8 Benachour S., Laurençot Ph., and Schmitt D., Extinction and decay estimates for viscous Hamilton-Jacobi equations in R^N, Proceedings of the American Mathematical Society. (2002) 130, no. 4, 1103–1111, https://doi.org/10.1090/S0002-9939-01-06140-8, MR1873785, ZBL1001.35007.
10.1090/S0002-9939-01-06140-8
Web of Science® Google Scholar
9 Benachour S., Laurençot Ph., Schmitt D., and Souplet Ph., Extinction and non-extinction for viscous Hamilton-Jacobi equations in R^N, Asymptotic Analysis. (2002) 31, no. 3-4, 229–246, MR1937838, ZBL1046.35053.
Web of Science® Google Scholar
10 Li Y. and Wu J., Extinction for fast diffusion equations with nonlinear sources, Electronic Journal of Differential Equations. (2005) 2005, no. 23, 1–7, MR2119075, ZBL1073.35134.
Web of Science® Google Scholar
11 Tian Y. and Mu C., Extinction and non-extinction for a p-Laplacian equation with nonlinear source, Nonlinear Analysis: Theory, Methods & Applications A. (2008) 69, no. 8, 2422–2431, https://doi.org/10.1016/j.na.2007.08.021, MR2446340.
10.1016/j.na.2007.08.021
Web of Science® Google Scholar
12 Liu W. and Wu B., A note on extinction for fast diffusive p-Laplacian with sources, Mathematical Methods in the Applied Sciences. (2008) 31, no. 12, 1383–1386, https://doi.org/10.1002/mma.976, MR2426994.
10.1002/mma.976
Web of Science® Google Scholar
13 Yin J. and Jin C., Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources, Mathematical Methods in the Applied Sciences. (2007) 30, no. 10, 1147–1167, https://doi.org/10.1002/mma.833, MR2329637.
10.1002/mma.833
Web of Science® Google Scholar
14 Leung A. W. and Zhang Q., Finite extinction time for nonlinear parabolic equations with nonlinear mixed boundary data, Nonlinear Analysis: Theory, Methods & Applications A. (1998) 31, no. 1-2, 1–13, https://doi.org/10.1016/S0362-546X(96)00191-5, MR1487527, ZBL0910.35069.
10.1016/S0362-546X(96)00191-5
Web of Science® Google Scholar

All articles

Extinction and Nonextinction for the Fast Diffusion Equation

Abstract

1. Introduction

2. Preliminaries

3. Main Results

3.1. Extinction of the Solution

3.2. Nonextinction of the Solution

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Extinction and Nonextinction for the Fast Diffusion Equation

Abstract

1. Introduction

2. Preliminaries

3. Main Results

3.1. Extinction of the Solution

3.2. Nonextinction of the Solution

Acknowledgments

References

References

Related

Information