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.
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.
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