Abstract
This article deals with the blowup problems of the positive solutions to a nonlinear parabolic equation with nonlocal source and nonlocal boundary condition. The blowup and global existence conditions are obtained. For some special case, we also give out the blowup rate estimate.
Keywords:
parabolic equation; nonlocal source; nonlocal nonlinear boundary condition; existence; blowup1. Introduction
In this article, we consider the positive solution of the following degenerate parabolic equation
where a, l > 0 and Ω is a bounded domain in R^{N }(N ≥ 1) with smooth boundary ∂Ω.
There have been many articles dealing with properties of solutions to degenerate parabolic equations with homogeneous Dirichlet boundary condition (see [14] and references therein). For example, Deng et al. [5] studied the parabolic equation with nonlocal source
which is subjected to homogeneous Dirichlet boundary condition. It was proved that there exists no global positive solution if and only if and , where φ(x) is the unique positive solution of the linear elliptic problem
However, there are some important phenomena formulated into parabolic equations which are coupled with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see [6,7]). Friedman [8] studied the problem of nonlocal boundary conditions for linear parabolic equations of the type
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
Parabolic equations with both nonlocal sources and nonlocal boundary conditions have been studied as well (see [912]). Lin and Liu [13] considered the problem of the form
They established local existence, global existence, and nonexistence of solutions, and discussed the blowup properties of solutions.
Chen and Liu [14] considered the following nonlinear parabolic equation with a localized reaction source and a weighted nonlocal boundary condition
Under certain conditions, they obtained blowup criteria. Furthermore, they derived the uniform blowup estimate for some special f(u).
In recent few years, reactiondiffusion problems coupled with nonlocal nonlinear boundary conditions have also been studied. Gladkov and Kim [15] considered the following problem for a single semilinear heat equation
where p, l > 0. They obtained some criteria for the existence of global solution as well as for the solution to blowup in finite time.
For other works on parabolic equations and systems with nonlocal nonlinear boundary conditions, we refer readers to [1620] and the references therein.
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 blowup or not of solutions.
In this article, we give the following hypotheses:
(H1) for α∈(0,1),u_{0}(x) > 0 in Ω, on ∂Ω.
(H2) g(x, y)≢0 is a nonnegative and continuous function defined for .
(H3) f∈C([0,∞))∩C^{1}(0,∞), f > 0, f' ≥ 0 in (0,∞).
The main results of this article are stated as follows.
Theorem 1.1. Assume that 0 < l ≤ 1 and for all x∈∂Ω.
(1) If a is sufficiently small, then the solution of (1.1) exists globally;
(2) 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 u_{0}(x) are sufficiently small. While the solution blows up in finite time if a,u_{0}(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 u_{0}(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 blowup of solutions, we need an additional assumption on the initial data u_{0}.
(H4) There exists a constant ε > ε_{1 }> 0 such that , where ε_{1 }will be given later.
Theorem 1.5. Assume u_{0}(x) satisfies (H1), (H2), and (H4), Δu_{0 }≤ 0 in Ω holds, and let f(u) = u^{p},0 < p ≤ 1, l = 1, then the following limits converge uniformly on any compact subset of Ω:
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 blowup for problem (1.1) are given in Section 3. In Section 4, the global blowup result and the blowup rate estimate of blowup 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 .
Definition 2.1. A function
 u
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 [10] and its proof is therefore omitted here.
Lemma 2.2. Suppose that and satisfies
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}, c_{5 }(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, c_{5 }(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 [21], the representation formula and the contraction mapping principle as in [13]. 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 blowup 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.
Proof of Theorem 1.1. (1) Let ψ(x) be the unique positive solution of the linear elliptic problem
where ε_{0 }is a positive constant such that 0 < ψ(x) < 1 (since , there exists such ε_{0}). Let , .
We define a function w(x, t) as following:
where M ≥ 1 is a constant to be determined later. Then, we have
On the other hand, we have for x ∈ Ω, t > 0,,
We choose and set , then it is easy to verify that w(x, t) is a supersolution of (1.1) provided that a ≤ a_{0}. By comparison principle, u(x, t) ≤ w(x, t), then u(x, t) exists globally.
(2) Consider the following problem
where , b_{1 }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) ≥ z_{0}.
Let v(x, t) = z (t) ψ (x), where ψ (x) is given by (3.1), then for x ∈ Ω, t > 0, we obtain
Set , if a is sufficiently large such that a > a_{1}, then we can choose
On the other hand, for x ∈ ∂Ω, t > 0, we get
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 u_{0 }(x) ≤ ψ (x) < 1 and , so the solution of (1.1) exists globally by using the comparison principle.
To prove the bowup result, we introduce the elliptic problem
Under the hypothesis (H2) and , we know that it exists a unique positive solution φ(x). Let , , and z(t) be the solution of the following ODE
Then, z(t) is increasing and z (t) ≥ z_{1}. Due to the condition for some positive constant δ, we know that z (t) of problem (3.10) blows up in finite time.
If a and u_{0 }(x) are so large that u_{0 }(x) ≥ z_{1 }(K*)^{l}, then we set v_{1}(x, t) = z(t)φ^{l}(x). For x ∈ Ω, t > 0, we obtain
For x ∈ ∂Ω, t > 0, by Jensen's inequality, we get
The inequalities (3.10)(3.13) show that v_{1}(x, t) is a subsolution of problem (1.1). Since v_{1}(x, t) blows up in finite time, u(x, t) also blows up in finite time by comparison principle.
Proof of Theorem 1.3. Let z(t) be the solution of the following ODE
where 0 < b_{2 }< a Ω. If u_{0}(x) is large enough, we can set . Then, z (t) is increasing and satisfies z (t) ≥ z_{2 }> 1. Moreover, z (t) of problem (3.14) blows up in finite time.
Set s (x, t) = z (t), then we have for x ∈ Ω, t > 0,
For x ∈ ∂Ω, t > 0,
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.
Proof of Theorem 1.4. Consider the following equation
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 [5], we can obtain the result immediately.
4. Blowup rate estimate
Now, we consider problem (1.1) with f (u) = u^{p}, 0 < p ≤ 1 and l = 1, i.e.,
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.
Lemma 4.1. Suppose that u_{0 }satisfies (H1), (H2), and (H4), then there exists a positive constant c_{0 }such that
Proof. By the first equation in (4.1), we have (see [22])
Hence,
Integrating (4.3) over (t, T*), we can get
Setting c_{0 }= (ap Ω p)^{1/p}, then we draw the conclusion.
Lemma 4.2. Under the conditions of Lemma 4.1, there exists a constant ε_{1}, which will be given below, such that
Proof. Let J(x, t) = u_{t}ε_{1}u^{p+1 }for (x, t) ∈Ω × (0, T*), a series of computations yields
By virtue of Hölder inequality, we have
Furthermore, by Young's inequality, for any θ > 0, the following inequality holds
Using (4.8) and taking , ε_{1 }= aΩ, then (4.7) becomes
Fix (x, t) ∈∂Ω × (0, T*), then we have
Since u_{t }(y, t) = J(y, t) + ε_{1}u^{p+1 }(y, t), we have
Noticing that p > 0, , we can apply Jensen's inequality to the last integral in the above inequality,
Hence, for (x, t) ∈∂Ω × (0, T*), we have
On the other hand, (H4) implies that
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 }≥ ε_{1}u^{p+1}. This completes the proof.
Integrating (4.6) from t to T*, we conclude that
where c_{2 }= (ε_{1}p)^{1/p }is a positive constant independent of t. Combining (4.2) with (4.12), we obtain the following result.
Theorem 4.3. Under the conditions of Lemma 4.1, if u (x, t) is the solution of (4.1) and blows up in finite time T*, then there exist positive constants c_{1}, c_{2}, such that
Lemma 4.4. Assume that u_{0}(x) satisfies (H1), (H2), and (H4), Δu_{0 }≤ 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 [14].
Denote
Lemma 4.5. Under the conditions of Lemma 4.4, it holds that
Proof. From Lemma 4.3, we have
Integrating (4.14) over (0, t), we obtain
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, .
Lemma 4. 6. Under the conditions of Lemma 4.4, then we have
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, .
In case of (1). Define z_{1}(x, t) = G (t)  u^{1p}/(1p), . A direct computation shows
where and using the equality . From (4.15), we know that
Integrate (4.20) from 0 to t,
Thus, (4.19) and (4.21) imply
Define K_{ρ }= {y∈ Ω: dist(y, ∂Ω)≥ ρ}. Since Δz_{1 }≤ 0 in Ω × (0, T*). Using Lemma 4.5 in [1], we obtain
It follows from (4.22) and (4.15) that
for any x∈ K_{ρ }and t ∈ (0,T*), where k_{1 }and K_{1 }are positive constants.
We know from Theorem 4.3 that
In view of (4.15) and Theorem 4.3, it follows that
From (4.23)(4.25), we get
It is obvious that
Thus,
In case of (2). We define z_{2 }(x, t) = G (t)  ln u(x, t), . Then,
From (4.16), we know that
Then,
Integrate (4.28) from 0 to t yields
Thus, (4.29) and (4.27) imply
Define K_{ζ }= {y∈ Ω: dist(y, ∂Ω)≥ ζ}. Since Δz_{2 }≤ 0 in Ω × (0, T*), we obtain
It follows from (4.30) and (4.27) that
for any x∈ K_{ζ }and t ∈ (0,T*).
By Theorem 4.3, we have
On the other hand, we know form (4.16) and Theorem 4.3 that
From (4.31)(4.33), we get
It is easy to derive
Thus,
This completes the proof.
Proof of Theorem 1.5. Case 1: 0 < p < 1. Form (4.17), we have
where the notation u ~ v means .
Furthermore,
Integrating (4.34) over (t, T*) yields
So, we can get our conclusion by using (4.17) and (4.35).
Case 2: p = 1. In this case, for any given σ: 0 < σ ≪ 1. By (4.18), there exists 0 < t_{0 }< T* such that
Therefore,
In view of the righthand side of the (4.36), we have
Integrating the above inequality from t to T* yields that
Namely,
Similar arguments to the lefthand side of (3.36) yield that
Consequently, (4.37) and (4.38) guarantee that for t_{0 }≤ t ≤ T*,
Letting σ → 0, we have
because of . Due to ln u(x, t) ~ G(t) uniformly on any compact subset of Ω, the proof is complete.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The main results in this article were derived by GZ and LT. All authors read and approved the final manuscript.
Acknowledgements
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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