Blow up problems for a degenerate parabolic equation with nonlocal source and nonlocal nonlinear boundary condition
© Zhong and Tian; licensee Springer. 2012
Received: 5 September 2011
Accepted: 18 April 2012
Published: 18 April 2012
This article deals with the blow-up problems of the positive solutions to a nonlinear parabolic equation with nonlocal source and nonlocal boundary condition. The blow-up and global existence conditions are obtained. For some special case, we also give out the blow-up rate estimate.
Keywordsparabolic equation nonlocal source nonlocal nonlinear boundary condition existence blow-up
where a, l > 0 and Ω is a bounded domain in R N (N ≥ 1) with smooth boundary ∂Ω.
with uniformly elliptic operator and c(x)≤ 0. It was proved that the unique solution of (1.4) tends to 0 monotonically and exponentially as t →+∞ provided that
They established local existence, global existence, and nonexistence of solutions, and discussed the blow-up properties of solutions.
Under certain conditions, they obtained blow-up criteria. Furthermore, they derived the uniform blow-up estimate for some special f(u).
where p, l > 0. They obtained some criteria for the existence of global solution as well as for the solution to blow-up in finite time.
Motivated by those of works above, we will study the problem (1.1) and want to understand how the function f(u) and the coefficient a, the weight function g(x, y) and the nonlinear term u l (y, t) in the boundary condition play substantial roles in determining blow-up or not of solutions.
In this article, we give the following hypotheses:
(H1) for α∈(0,1),u0(x) > 0 in Ω, on ∂Ω.
(H2) g(x, y)≢0 is a nonnegative and continuous function defined for .
(H3) f∈C([0,∞))∩C1(0,∞), f > 0, f' ≥ 0 in (0,∞).
The main results of this article are stated as follows.
If a is sufficiently small, then the solution of (1.1) exists globally;
If a is sufficiently large, then the solution of (1.1) also exists globally provided that for some δ > 0.
Theorem 1.2. Assume that l > 1 and for all x∈∂Ω. Then the solution of (1.1) exists globally provided that a and u0(x) are sufficiently small. While the solution blows up in finite time if a,u0(x) are sufficiently large and for some δ > 0.
Theorem 1.3. Assume that l > 1 and for all x∈∂Ω. If for some δ > 0, then the solution of (1.1) blows up in finite time provided that u0(x) is large enough.
Theorem 1.4. If for some δ > 0 and , where φ(x) is the solution of (1.3), then there exists no global positive solution of (1.1).
To describe conditions for blow-up of solutions, we need an additional assumption on the initial data u 0 .
(H4) There exists a constant ε > ε1 > 0 such that , where ε1 will be given later.
If 0 < p < 1, .
If p = 1, .
This article is organized as follows. In Section 2, we establish the comparison principle and the local existence. Some criteria regarding to global existence and finite time blow-up for problem (1.1) are given in Section 3. In Section 4, the global blow-up result and the blow-up rate estimate of blow-up solutions for the special case of f (u) = u p , 0 < p ≤ 1 and l = 1 are obtained.
2. Comparison principle and local existence
First, we start with the definition of subsolution and supersolution of (1.1) and comparison principle. Let Q T = Ω × (0, T), S T = ∂Ω × (0, T), and .
Similarly, a supersolution ū(x, t) of (1.1) is defined by the opposite inequalities.
A solution of problem (1.1) is a function which is both a subsolution and a supersolution of problem (1.1).
The following comparison principle plays a crucial role in our proofs which can be obtained by similar arguments as  and its proof is therefore omitted here.
where d(x, t), c i (x, t)(i = 1,2,3,4) are bounded functions and d(x, t)≥ 0, c i (x, t)≥ 0 (i = 2,3,4) in Q T , c5 (x, y)≥ 0 for x∈∂Ω, y∈Ω and is not identically zero. Then, w(x, 0) > 0 for implies w(x, t) > 0 in Q T . Moreover, c5 (x, y) ≡ 0 or if on S T , then w(x, 0) ≥ 0 for implies w(x, t) ≥ 0 in Q T .
On the basis of the above lemmas, we obtain the following comparison principle of (1.1).
Lemma 2.3. Let u and v be nonnegative subsolution and supersolution of (1.1), respectively, with u(x, 0) ≤ v(x, 0) for . Then, u ≤ v in Q T if u ≥ η or v ≥ η for some small positive constant η holds.
Local in time existence of positive classical solutions of (1.1) can be obtained by using fixed point theorem , the representation formula and the contraction mapping principle as in . By the above comparison principle, we get the uniqueness of solution to the problem. The proof is more or less standard, so is omitted here.
3. Global existence and blow-up in finite time
In this section, we will use super- and subsolution techniques to derive some conditions on the existence or nonexistence of global solution.
where ε0 is a positive constant such that 0 < ψ(x) < 1 (since , there exists such ε0). Let , .
- (2)Consider the following problem(3.5)
where , b1 is a positive constant to be fixed later. It follows from hypothesis (H3) and the theory of ordinary differential equation (ODE) that there exists a unique solution z (t) to problem (3.5) and z (t) is increasing. If for some positive δ, we know that z (t) exists globally and z (t) ≥ z0.
Set , if a is sufficiently large such that a > a1, then we can choose
Here, we use the conclusions 0 < ψ (x) < 1 and z(t) > 1.
And the inequalities (3.5)-(3.9) show that v(x, t) is a supersolution of (1.1). Again by using the comparison principle, we obtain the global existence of u(x, t). The proof is complete.
Proof of Theorem 1.2. The proof of global existence part is similar to the first case of Theorem 1.1. For any given positive constant M ≤ 1, w (x) = Mψ (x) is a supersolution of problem (1.1) provided that u0 (x) ≤ ψ (x) < 1 and , so the solution of (1.1) exists globally by using the comparison principle.
Then, z(t) is increasing and z (t) ≥ z1. Due to the condition for some positive constant δ, we know that z (t) of problem (3.10) blows up in finite time.
The inequalities (3.10)-(3.13) show that v1(x, t) is a subsolution of problem (1.1). Since v1(x, t) blows up in finite time, u(x, t) also blows up in finite time by comparison principle.
where 0 < b2 < a |Ω|. If u0(x) is large enough, we can set . Then, z (t) is increasing and satisfies z (t) ≥ z2 > 1. Moreover, z (t) of problem (3.14) blows up in finite time.
From (3.14)-(3.17), we see that s (x, t) is a subsolution of (1.1). Hence, u (x, t) ≥ s (x, t) by comparison principle, which implies u (x, t) blows up in finite time. This completes the proof.
and let v (x, t) be the solution to problem (3.18). It is obvious that v (x, t) is a subsolution of (1.1). By Theorem 1 in , we can obtain the result immediately.
4. Blow-up rate estimate
where for all x ∈ ∂Ω, and suppose that the solution of (4.1) blows up in finite time T*.
Set , then U(t) is Lipschitz continuous.
Setting c0 = (ap |Ω| p)-1/p, then we draw the conclusion.
Owing to u(x, t) is a positive continuous function for , it follows from (4.9)-(4.11) and Lemma 2.2 that J(x, t) ≥ 0 for , i.e., u t ≥ ε1up+1. This completes the proof.
where c2 = (ε1p)-1/pis a positive constant independent of t. Combining (4.2) with (4.12), we obtain the following result.
Lemma 4.4. Assume that u0(x) satisfies (H1), (H2), and (H4), Δu0 ≤ 0 in Ω. u(x, t) is the solution of problem (4.1). Then, Δu ≤ 0 in any compact subsets of Ω × (0, T*).
The proof is similar to that of Lemma 1.1 in .
In view of , . Noting that u t ≥ 0 by the assumption of the initial function, then we see that g(t) is monotone nondecreasing. Therefore, .
uniformly on any compact subsets of Ω.
Proof. Let λ > 0 be the principal eigenvalue of -Δ in Ω with the null Dirichlet boundary condition, and ϕ(x) be the corresponding eigenfunction satisfying ϕ(x) > 0, .
for any x∈ K ρ and t ∈ (0,T*), where k1 and K1 are positive constants.
for any x∈ K ζ and t ∈ (0,T*).
This completes the proof.
where the notation u ~ v means .
So, we can get our conclusion by using (4.17) and (4.35).
because of . Due to ln u(x, t) ~ G(t) uniformly on any compact subset of Ω, the proof is complete.
The authors express their thanks to the referee for his or her helpful comments and suggestions on the manuscript of this article.
- Souple P: Uniform blow-up profiles and boundary for diffusion equations with nonlocal nonlinear source. J Diff Equ 1999, 153: 374-406. 10.1006/jdeq.1998.3535View ArticleGoogle Scholar
- Weissler FB: An L∞blow-up estimate for a nonlinear heat equation. Commun Pure Appl Math 1985, 38: 291-296. 10.1002/cpa.3160380303MathSciNetView ArticleGoogle Scholar
- Galaktionov VA: On asymptotic self-similar behavior for a quasilinear heat equation: single point blow-up. SIAM J Math Anal 1995, 26: 675-693. 10.1137/S0036141093223419MathSciNetView ArticleGoogle Scholar
- Giga Y, Umeda N: Blow-up directions at space infinity for solutions of semilinear heat equations. Bol Soc Parana Mat 2005, 23: 9-28.MathSciNetGoogle Scholar
- Deng W, Li Y, Xie C: Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations. Appl Math Lett 2003, 16: 803-808. 10.1016/S0893-9659(03)80118-0MathSciNetView ArticleGoogle Scholar
- Day WA: Extensions of property of heat equation to linear thermoelasticity and other theories. Q Appl Math 1982, 40: 319-330.Google Scholar
- Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Q Appl Math 1983, 40: 468-475.Google Scholar
- Friedman A: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Q Appl Math 1986, 44: 401-407.Google Scholar
- Cui ZJ, Yang ZD: Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition. J Math Anal Appl 2008, 342: 559-570. 10.1016/j.jmaa.2007.11.055MathSciNetView ArticleGoogle Scholar
- Deng K: Comparison principle for some nonlocal problems. Q Appl Math 1992, 50: 517-522.Google Scholar
- Pao CV: Asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions. J Comput Appl Math 1998, 88: 225-238. 10.1016/S0377-0427(97)00215-XMathSciNetView ArticleGoogle Scholar
- Wang Y, Mu C, Xiang Z: Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary. Boundary Value Problems Article ID 64579 2007, 1-12.Google Scholar
- Lin Z, Liu Y: Uniform blowup profiles for diffusion equations with nonlocal source and nonlocal boundary. Acta Mathematica Scientia (Series B) 2004, 24: 443-450.Google Scholar
- Chen YP, Liu LH: Global blow-up for a localized nonlinear parabolic equation with a nonlocal boundary condition. J Math Anal Appl 2011, 384: 421-430. 10.1016/j.jmaa.2011.05.079MathSciNetView ArticleGoogle Scholar
- Gladkov A, Kim KI: Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition. J Math Anal Appl 2008, 338: 264-273. 10.1016/j.jmaa.2007.05.028MathSciNetView ArticleGoogle Scholar
- Gladkov A, Kim KI: Uniqueness and nonuniqueness for reaction-diffusion equation with nonlinear nonlocal boundary condition. Adv Math Sci Appl 2009, 19: 39-49.MathSciNetGoogle Scholar
- Gladkov A, Guedda M: Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition. Nonlinear Anal 2011, 74: 4573-4580. 10.1016/j.na.2011.04.027MathSciNetView ArticleGoogle Scholar
- Liang J, Wang HY, Xiao TJ: On a comparison principle for delay coupled systems with nonlocal and nonlinear boundary conditions. Nonlinear Anal 2009, 71: 359-365. 10.1016/j.na.2008.11.012MathSciNetView ArticleGoogle Scholar
- Liu DM, Mu CL: Blowup properties for a semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions. Abstr Appl Anal Article ID 148035 2010, 1-17.Google Scholar
- Mu CL, Liu DM, Zhou SM: Properties of positive solutions for a nonlocal reaction diffusion equation with nonlocal nonlinear boundary condition. J Kor Math Soc 2011, 47: 1317-1328.MathSciNetView ArticleGoogle Scholar
- Yin HM: On a class of parabolic equations with nonlocal boundary conditions. J Math Anal Appl 2004, 294: 712-728. 10.1016/j.jmaa.2004.03.021MathSciNetView ArticleGoogle Scholar
- Friedman A, Mcleod J: Blow-up of positive solutions of semilinear heat equations. Indiana Univ Math 1985, 34: 425-447. 10.1512/iumj.1985.34.34025MathSciNetView ArticleGoogle Scholar