Abstract
In this article, we investigate the Dirichlet problem for a porous medium equation with a more complicated source term. In some cases, we prove that the solutions have global blowup and the rate of blowup is uniform in all compact subsets of the domain. Moreover, in each case, the blowup rate of u(t)_{∞ }is precisely determined.
Keywords:
porous medium equation; localized source; blowup, uniform blowup rate1 Introduction
Let Ω be a bounded domain in ℝ^{N }(N ≥ 1) with smooth boundary ∂Ω. We consider the following parabolic equation with a localized reaction term
where m ≥ 1, q_{1 }≥ 0, s_{1 }> 0 and x_{0 }∈ Ω is a fixed point. Throughout this article, we assume the functions a(x) and v_{0}(x) satisfy the following conditions:
(A1) a(x) and v_{0}(x) ∈ C^{2}(Ω); a(x), v_{0}(x) > 0 in Ω and a(x) = v_{0}(x) = 0 on ∂Ω.
When Ω = B = {x ∈ ℝ^{N}; x < R}, we sometimes assume
(A2) a(x) and v_{0}(x) are radially symmetric; a(r) and v_{0}(r) are nonincreasing for r ∈ [0, R].
Problems (1.1)(1.3) arise in the study of the flow of a fluid through a porous medium with an internal localized source and in the study of population dynamics (see [13]). Porous medium equations (m > 1) with or without local sources have been studied by many authors [46].
Concerning (1.1)(1.3), to the best of authors knowledge, a number of articles have studied it from the point of the view of blowup and global existence [710]. Many studies have been devoted to the case m = 1 [1013]. The case m = 1, a(x) = 1, q_{1 }= 0, s_{1 }≥ 1 and m = 1, a(x) = 1, q_{1}, s_{1 }> 1 were studied by Souple [10,11]. Souple [10] demonstrated that the positive solution blows up in finite time if the initial value v_{0 }is large enough. In the case a(x) = 1, q_{1 }= 0, and s_{1 }> 1, Souple [11] showed that the solution v(x, τ) blows up globally and the blowup rate is precisely determined. The case q_{1 }= 0 and s_{1 }> 0 was studied by Cannon and Yin [12] and Chandam et al. [13]. Cannon and Yin [12] studied its local solvability and Chandam et al. [13] investigated its blowup properties.
The study of this article is motivated by some recent results of related problems (see [14][15][16]. In the case of a(x)(= constant), the global existence and blowup behavior have been considered by Chen and Xie [15]. It turns out that if q_{1 }+ s_{1 }< m or q_{1 }+ s_{1 }= m and a(x)(= constant) is sufficiently small, there exists a global solution of problem (1.1)(1.3); if q_{1 }+ s_{1 }> m, the solution of problem (1.1)(1.3) blows up for large initial datum while it admits a global solution for small initial datum. Furthermore, Du and Xiang [16] obtained the blowup rate estimates under some appropriate hypotheses on initial datum. For some related localized models arising in physical phenomena, we refer the readers to [1719] and the references therein.
For the localized semilinear parabolic equation of the form
with the Dirichlet boundary condition (1.2) and the initial condition (1.3). In [20], Li and Wang proved that the blowup set to system (1.2)(1.4): (a) the system possesses total blowup when q_{1 }≤ 1; (b) the system presents single point blowup patterns when q_{1 }> 1.
We now restrict ourselves to the problem of the form
where q_{1 }≥ 0, s_{1 }> 0, and q_{1 }+ s_{1 }> m > 0. When m = 1, it was proved in [14] that
(1) If 0 ≤ q_{1 }≤ 1 and q_{1 }+ s_{1 }> 1, then the solution of (1.5)(1.7) blows up in a finite time T.
(2) If q_{1 }> 1, then x = 0 is the only blowup point for (1.5)(1.7).
In the meantime, they obtained the blowup rate estimate but less precise. Namely,
(i) If 0 ≤ q_{1 }< 1, then for any x ∈ B
(ii) If q_{1 }= 1, then for any x ∈ B
It seems that the results of [14] can be extended to m ≥ 1 and the blowup rate can be precisely determined. Motivated by this, in this article, we will extend and improve the results of [14].
The purpose of this article is to determine the blowup rate of solutions for a nonlinear parabolic equation with a weighted localized source, that is, we investigate how the localized source and the local term affect the blowup properties of the problem (1.5)(1.7). Indeed, we find that when q_{1 }≤ 1, the solution of (1.5)(1.7) blows up at the whole domain with a uniformly blowup profile.
The rest of this article is organized as follows. The results are stated in Section 2. We then prove these results in Section 3.
2 Preliminaries and Main Results
The following two theorems are our main results.
Theorem 2.1 Assume q_{1 }+ s_{1 }> m, (A1) and (A2) hold. Let v(x, t) be the solution of problem (1.5)(1.7), then v(x,t) blows up provided that the initial value v_{0}(x) is sufficiently large.
The method used in the proof Theorem 2.1 is originally due to [8,18], and bears much resemblance to that of Theorem 3.2 in [15] and Theorem 1.3 in [16]. Therefore, we omitted them here.
For the case q_{1 }> 1, we do not know how to deal with the uniform blowup rate of problem (1.5)(1.7). In the following, we focus only on the case of 0 ≤ q_{1 }≤ 1.
Theorem 2.2 Assume (A1) and (A2) hold. Let v(x, t) be the blowup solution of (1.5)(1.7), which blows up in finite time T and v(x, t) is nondecreasing in time, then the following limits hold uniformly in all compact subsets of B.
(i) If 0 ≤ q_{1 }< 1, then
(ii) If q_{1 }= 1, then
Remark 2.1 The domain we considered here is a ball, it seems that the results of Theorem 2.2 remain valid for the general domain. (It is an open problem in this case.)
To get the blowup profiles for problem (1.5)(1.7), we need some transformations. Let u(x, t) = v^{m}(x, τ), t = mτ, then (1.5)(1.7) becomes
where 0 ≤ p = (m  1)/m < 1, q = q_{1}/m, and s = s_{1}/m.
Under above transformation, assumptions (A1) and (A2) become
(B1) a(x) and u_{0}(x) ∈ C^{2}(B); a(x), u_{0}(x) > 0 in B and a(x) = u_{0}(x) = 0 on ∂B.
(B2) a(x) and u_{0}(x) are radially symmetric; a(r) and u_{0}(r) are nonincreasing for r ∈ [0, R].
In our consideration, a crucial role is played by the Dirichlet eigenvalue problem
Denote λ be the first eigenvalue and by φ the corresponding eigenfunction with φ(x) > 0 in B, normalized by .
3 Proof of Theorem 2.2
For convenience, we denote
Before proving our result, we would like to give a property of the following problem
where 0 ≤ α ≤ 1 and w = u^{1q}(x, t).
Lemma 3.1 Assume (B1) and (B2) hold. Let w(x, t) be the solution of Equation (3.1), which blows up in a finite time T* and nondecreasing in time t, then the following limits hold uniformly in all compact subsets of B.
(i) If 0 ≤ α < 1, then
(ii) If α = 1, then
Proof. (i) Assumption (B2) implies w_{r }≤ 0 (r = x), it then follows that and Δw(0, t) ≤ 0 for t > 0. From (3.1), we then get
.
Consequently,
which implies
Moreover, it is apparent that lim_{tT* }w(0, t)/g(t) = 0, since s > 1  q.
Set R_{1 }∈ (0, R), B_{1 }= {x ∈ ℝ^{N},  x < R_{1}} and b(x) = 1/a(x), x ∈ B_{1}. Since a^{'}(r) ≤ 0, we obtain that b'(r) ≥ 0, for 0 ≤ r ≤ R_{1}.
We now introduce the function
By a simple calculation, and note that ∇w(x, t)∇b(x) = u_{r}(r, t)b'(r) ≤ 0, then there exist m_{1}, m_{2 }> 0 such that
Setting ε(t) = m_{2}w(0, t)/g(t). From lim_{t→T* }w(0, t)/g(t) = 0, we infer that there exists t_{1 }∈ (0, T*) such that 0 < ε(t) ≤ 1/2 for t_{1 }≤ t < T*.
Hence, in view of (3.1), we observe
Set g_{1}(t) = (1  ε(t))g(t), , we then obtain
Obviously, w_{1}(x, t) is a supsolution of the following equation
By the maximum principle, w_{1}(x, t) ≥ w*(x, t) and . Similar to the proof of (4.15) in [15] that
uniformly in all compact subsets of B_{1},
Therefore, by the arbitrariness of B_{1}, we obtain that the following inequatlity holds uniformly in all compact subsets of B
In particular,
From (3.2) and (3.4), we deduce
Multiplying both sides of (3.1) by φ and integrating over B × (0, t), 0 < t < T*
It then follows that
Note that w_{r }≤ 0, (3.3) and (3.6), it is sufficient to prove
Assume on the contrary that there exists a point x_{1 }∈ B, x_{1 }≠ 0 such that
Then there exists a sequence {t_{n}} such that t_{n }→ T*
By the continuity of a(x), we deduce that there exists x_{2 }∈ B such that (1  α)a(x) < c for B_{1 }= {x ∈ ℝ^{n }: x_{2} ≤ x ≤ x_{1}}. Using w_{r }≤ 0, (3.3) and (3.6), it is easy to check that
which is a contradiction to (3.6). Combining (3.3) and (3.7), Lemma 3.1 (i) is proved. Case (ii) can be treated similarly.
The key step in establishing the result of Theorem 2.2 is the following lemma.
Lemma 3.2 Under the assumption of Lemma 3.1, let u(x, t) be the blowup solution of (2.3), which blows up in a finite time T* and nondecreasing in time t, then the following statements hold uniformly in all compact subsets of B:
(i) If p + q < 1, then
(ii) If p + q = 1, then
Proof. (i) Since u_{r }≤ 0 and u_{t }≥ 0, it then follows that and Δu(0, t) ≤ 0 for t > 0, which imply lim_{t→T* }u(0, t) = ∞. Obviously,
which implies
Notice that p + q < 1 and (3.8), hence lim_{t→T* }G(t) = ∞ and lim_{t→T* }g(t) = ∞.
A simple calculation yields
In view of (2.3), we have, for x ∈ Ω, 0 < t < T*
Multiplying both sides of Equation (3.9) by φ and integrating over B × (0, t), it follows that
for 0 < t < T*. Clearly,
which yields
Setting u_{1}(r, t) = u^{(1q)/2}(r, t)(r = x). We may claim that
Indeed, due to lim_{t→T* }g(t) = lim_{t→T* }u^{s}(0, t) = ∞, u_{r }≤ 0, and s > 1  q, we then have
Therefore, by Lebesgue's dominated convergence theorem, we infer that
where w_{n }is the surface area of unit ball in ℝ^{N}.
Now according to (3.10)(3.12), we obtain
On the other hand, By (3.9), we find
where γ = p/(1  q). Consequently, u^{1q }is a supsolution of the problem
By the maximum principle, u^{1q }≥ v in B × (0, T*). Note that 0 ≤ γ < 1, we know from Lemma 3.1 (i) that
uniformly in all compact subsets of B.
Thus,
uniformly in all compact subsets of B.
Next, we prove that
uniformly in all compact subsets of B.
We can verify (3.15) by similar means of (3.7). Therefore, we conclude the proof of case (i).
(ii) Proceeding as (3.8), we have
For any compact subset B_{1 }∈ B, there exists t_{1 }∈ (0, T*) such that u(x, t_{1}) ≥ 1 for all , and thus ln u(x, t) ≥ 0 in .
Direct calculation shows
Let λ_{1 }be the first eigenvalue of Δ in and by φ_{1 }> 0 the corresponding eigenfunction, normalized by . Set . Clearly, lim_{t→T* }G(t)/G_{1}(t) = 1.
Multiplying both sides of Equation (3.16) by φ_{1 }and integrating over B_{1 }× (t_{1}, t), we get
The result of case (ii) follows by analogy with the argument used in the proof of case (i).
Proof of Theorem 2.2
(i) By Lemma 3.2 (i), we infer that
hence
Integrating equivalence (3.18) between t and T*, we obtain
Using Lemma 3.2 (i) and substituting p = (m  1)/m, q = q_{1}/m, s = s_{1}/m, t* = mτ, and u(x, t) = v^{m}(x, τ) into (3.19), we complete the proof of Theorem 2.2 (i).
(ii) To obtain the blowup rate of the exponent type, we need to be more careful in this case, since exponentiation of equivalents is not permitted. Similar to the proof of Theorem 3 in [14] and Lemma 2.3 in [16], we get
Thanks to Lemma 3.2(ii) and (3.20), we then get the desired result.
4 Discussion
This article deals with the porous medium equation with local and localized source terms, represented by two factors and , respectively. As we all know that, in the absence of weight function, the solutions of model (1.5)(1.7) have a global blowup and the rate of blowup is uniform in all compact subsets of the domain. A natural question is what happens in the model (1.5)(1.7), where the source term is the product of localized source, local source, and weight function. It is shown by Theorem 2.2 that if 0 ≤ q_{1 }≤ 1, this equation possesses uniform blowup profiles. In other words, the localized term plays a leading role in the blowup profile for this case. Moreover, the blowup rate estimates in time and space is obtained.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All the authors typed, read, and approved the final manuscript.
Acknowledgements
The authors thank the anonymous referee for their constructive and valuable comments, which helped in improving the presentation of this study. This study was supported by the National Natural Science Foundation of China (60972001), the National Key Technologies R & D Program of China (2009BAG13A06), the Scientific Innovation Research of College Graduate in Jiangsu Province (CXZZ_0163), and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1140).
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