Abstract
This paper investigates the blowup properties of positive solutions to the following system of evolution pLaplace equations with nonlocal sources and inner absorptions
with homogeneous Dirichlet boundary conditions in a smooth bounded domain Ω ∈ R^{N}(N ≥ 1), where p, q > 2, m, n, r, s ≥ 1, α, β > 0. Under appropriate hypotheses, the authors discuss the global existence and blowup of positive weak solutions by using a comparison principle.
2010 Mathematics Subject Classification: 35B35; 35K60; 35K65; 35K57.
Keywords:
evolution pLaplace system; global existence; blowup; nonlocal sources; absorptions1 Introduction
In this paper, we deal with the blowup properties of positive solutions to an evolution pLaplace system of the form
where p, q > 2, m, n, r, s ≥ 1, α, β > 0, Ω is a bounded domain in R^{N}(N ≥ 1) with a smooth boundary ∂Ω, the initial data , and , , where v denotes the unit outer normal vector on ∂Ω.
System (1.1) is the classical reactiondiffusion system of Fujitatype for p = q = 2. If p ≠ 2, q ≠ 2, (1.1) appears in the theory of nonNewtonian fluids [1,2] and in nonlinear filtration theory [3]. In the nonNewtonian fluids theory, the pair (p, q) is a characteristic quantity of the medium. Media with (p, q) > (2, 2) are called dilatant fluids and those with (p, q) < (2, 2) are called pseudoplastics. If (p, q) = (2, 2), they are Newtonian fluids.
System (1.1) has been studied by many authors. For p = q = 2, Escobedo and Herrero [4] considered the following problem
where p, q > 0. Their main results read as follows. (i) If pq ≤ 1, every solution of (1.2) is global in time. (ii) If pq > 1, some solutions are global while some others blow up in finite time.
In the last three decades, many authors studied the following degenerate parabolic problem
under different conditions (see [5,6] for nonlinear boundary conditions; see [710] for local nonlinear reaction terms; see [11] for nonlocal nonlinear reaction terms). In [12], the existence, uniqueness, and regularity of solutions were obtained. When f(u) = u^{q}, q > 0 or f(u) ≡ 0 extinction phenomenon of the solution may appear [1315]; However, if f(u) = u^{q}, q > 1 the solution may blow up in finite time [710,14].
Especially, in [11], Li and Xie dealt with the following pLaplace equation
Under appropriate hypotheses, they established the local existence and uniqueness of its solution. Furthermore, they obtained that the solution u exists globally if q < p  1; u blows up in finite time if q > p  1 and u_{0}(x) is large enough.
Recently, in [16], Li generalized (1.4) to system and studied the following problem
Similar to [11], he proved that whether the solution blows up in finite time depends on the initial data, constants α, β, and the relations between mn and (p  1)(q  1).
For other works on parabolic system like (1.1), we refer readers to [1730] and the references therein.
When p = q, m = n, r = s, α = β, u_{0}(x) = v_{0}(x), system (1.1) is then reduced to a single pLaplace equation
However, to the authors' best knowledge, there is little literature on the study of the global existence and blowup properties for problems (1.1) and (1.6). Motivated by the above works, in this paper, we investigate the blowup properties of solutions of the problem (1.1) and extend the results of [4,11,16,19] to more generalized cases.
In order to state our results, we introduce some useful symbols. Throughout this paper, we let φ(x), ψ(x) be the unique solution of the following elliptic problem
and
respectively. For convenience, we denote
Before starting the main results, we introduce a pair of parameters (μ, γ) solving the following characteristic algebraic system
namely,
with
It is obvious that 1/τ and 1/θ share the same signs. We claim that the critical exponent of problem (1.1) should be (1/τ, 1/θ) = (0, 0), described by the following theorems.
Theorem 1.1. Assume that (1/τ, 1/θ) < (0, 0), then there exist solutions of (1.1) being globally bounded.
Theorem 1.2. Assume that (1/τ, 1/θ) > (0, 0), then the nonnegative solution of (1.1) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small initial values.
Theorem 1.3. Assume that (1/τ, 1/θ) = (0, 0), φ(x) and ψ(x) are defined in (1.7) and (1.8), respectively.
(i) Suppose that r > p  1 and s > q  1. If α^{n}β^{r }≥ Ω^{n+r}, then the solutions are globally bounded for small initial data; if , , then the solutions blow up in finite time for large data.
(ii) Suppose that p  1 > r and q  1 > s. If , then the solutions are globally bounded for small initial data; if , then the solutions blow up in finite time for large data.
(iii) Suppose that p  1 > r and s > q  1. If , then the solutions are globally bounded for small initial data; if , , then the solutions blow up in finite time for large data.
(iv) Suppose that r > p  1 and q  1 > s. If , then the solutions are globally bounded for small initial data; if , , then the solutions blow up in finite time for sufficiently large data.
The rest of this paper is organized as follows. In Section 2, we shall establish the comparison principle and local existence theorem for problem (1.1). Theorems 1.1 and 1.2 will be proved in Section 3 and Section 4, respectively. Finally, we will give the proof of Theorem 1.3 in Section 5.
2 Preliminaries
Since the equations in (1.1) are degenerate at points where ∇u = 0 or ∇v = 0, there is no classical solution in general, and we therefore consider its weak solutions. Let Ω_{T }= Ω × (0, T), S_{T }= ∂Ω × (0, T) and . We begin with the precise definition of a weak solution of problem (1.1).
Definition 2.1 A pair of functions (u(x, t), v(x, t)) is called a weak solution of problem (1.1) in if and only if
(i) (u, v) is in the space and (u_{t}, v_{t}) ∈ L^{2}(0, T; L^{2}(Ω)) × L^{2}(0, T; L^{2}(Ω)).
(ii) the following equalities
and
hold for all ϕ_{1}, ϕ_{2}, which belong to the class of test functions
(iii) u(x, t)_{t = 0 }= u_{0}(x), v(x, t)_{t = 0 }= v_{0}(x) for all .
In a natural way, the notion of a weak subsolution for (1.1) is given as follows.
Definition 2.2 A pair of functions (
 u
 v
(i) (
 u
 v
 u
 v
(ii) the following inequalities
and
hold for any ϕ_{1}, ϕ_{2}, which belong to the class of test functions
(iii)
 u
 v
Similarly, a pair of functions is a weak supersolution of (1.1) if the reversed inequalities hold in Definition 2.2. A weak solution of (1.1) is both a weak subsolution and a weak supersolution of (1.1).
We shall use the following comparison principle to prove our global and nonglobal existence results.
Proposition 2.3 Let (
 u
 v
Proof. From the definitions of weak subsolution and supersolution, for any ϕ_{1}, ϕ_{2 }∈ Θ_{2}, we could obtain that
and
In addition, inequalities (2.1) and (2.2) remain true for any subcylinder of the form Ω_{τ }= Ω × (0, τ) ⊂ Ω_{T }and corresponding lateral boundary S_{τ }= ∂Ω × (0, τ) ⊂ S_{T}. Taking a special test function in (2.1), where χ_{[0, τ] }is the characteristic function defined on [0, τ] and s_{+ }= max{s, 0}, we find that
where Ω denotes the Lebesgue measure of Ω and
Next, our task is to estimate the first term on the rightside of (2.3). In view of Cauchy's inequality, we see that
Furthermore, by Lemma 1.4.4 in [12], we know that there exists δ > 0 such that
Combining now (2.3)(2.5), we deduce that
Likewise, taking test function in (2.2), we have that
where C_{3}, C_{4 }denote some positive constants. Moreover, there exists a large enough constant C, such that
Now, we write
then, (2.8) implies that
By Gronwall's inequality, we know that y(τ) = 0, for any τ ∈ [0, T]. Thus, , this means that , in as desired. The proof of Proposition 2.3 is complete. □
With the above established comparison principle in hand, we are able to show the basic existence theorem of weak solutions. Here, we only state the local existence theorem, and its proof is standard [12, 16, for more details].
Theorem 2.1 Given , there is some T_{0 }> 0 such that the problem (1.1) admits a nonnegative unique weak solution (u, v) for each t < T_{0}, and . Furthermore, either T_{0 }= ∞ or
3 Proof of Theorem 1.1
Proof of Theorem 1.1. Notice that (1/τ, 1/θ) < (0, 0) implies
We will prove Theorem 1.1 in four subcases.
(a) For μ = r, γ = s, we then have mn < rs. Let , where , will be determined later. After a simple computation, we have
and
So, is a timeindependent supersolution of problem (1.1) if
i.e.,
(b) For μ = p  1, γ = q  1, we then have mn < (p  1)(q  1). Let
where φ, ψ satisfying (1.7) and (1.8), respectively. Taking
and
then it is easy to verify that is a global supersolution for system (1.1).
(c) For μ = r, γ = q  1, we then have mn < r(q  1). Choose and satisfy
Let with ψ defined by (1.8). By direct Computation, we arrive at
and
(d) For μ = p  1, γ = s, we then have mn < r(q  1). Let with φ defined by (1.7), where and . Then, (3.2) and (3.3) hold if
The proof of Theorem 1.1 is complete. □
4 Proof of Theorem 1.2
Proof of Theorem 1.2. Observe that 1/τ, 1/θ > 0 implies
For μ = r, γ = s. Choosing
then is a global supersolution for problem (1.1) provided that and .
For μ = p  1, γ = q  1. Let , where φ and ψ satisfying (1.7) and (1.8), respectively. Choosing
and
therefore, is a global supersolution for system (1.1) if and .
For other cases, the solutions of (1.1) should be global due to the above discussion.
Next, we begin to prove our blowup conclusion under large enough initial data. Due to the requirement of the comparison principle, we will construct blowup subsolutions in some subdomain of Ω in which u, v > 0. We use an idea from Souplet [31] and apply it to degenerate equations. Since problem (1.1) does not make sense for negative values of (u, v), we actually consider the following problem
where u_{+ }= max{0, u}, v_{+ }= max{0, v}. Let ϖ(x) be a nontrivial nonnegative continuous function and vanish on ∂Ω. Without loss of generality, we may assume that 0 ∈ Ω and ϖ(0) > 0. We shall construct a selfsimilar blowup subsolution to complete our proof.
Set
here
and l_{i}, σ_{i }> 0(i = 1, 2), 0 < T < 1 are to be determined later. Notice the fact that
for sufficiently small T > 0.
Calculating directly, we obtain
and
where
On the other hand, we know
here H_{x}(
 u
 v
 u
 v
Further, we have
and
Since 1/τ, 1/θ < 0, we see that μγ < mn. In addition, it is clear that
For , we choose l_{1 }and l_{2 }such that
Recall that μ = max{p  1, r} and γ = max{q  1, s}, then (4.9) implies
and
Next, we can choose positive constants σ_{1}, σ_{2 }sufficiently small such that
consequently, we have
For , we fix l_{1 }and l_{2 }to satisfy
then we can also select σ_{1}, σ_{2 }small enough such that (4.10) holds.
From (4.6), (4.7) and (4.10), for sufficiently small T > 0, it follows that
Since ϖ(0) > 0 and ϖ(x) are continuous, there exist two positive constants ρ and ε such that ϖ(x) ≥ ε for all x ∈ B(0, ρ) ⊂ Ω. Choose T small enough to insure , hence
 u
 v
 u
 v
5 Proof of Theorem 1.3
Proof of Theorem 1.3. In the critical case of (1/τ, 1/θ) = (0, 0), we have mn = μγ.
(i) For r > p  1, s > q  1, we know mn = rs. Thanks to α^{n}β^{r }≥ Ω^{n+r}, we can choose A and B sufficiently large such that , and
Clearly, is a supersolution of problem (1.1), then by comparison principle, the solution of (1.1) should be global.
Next, we begin to prove our blowup conclusion. Since mn = rs, we can choose constants l_{1}, l_{2 }> 1 such that
According to Proposition 2.3, we only need to construct a suitable blowup subsolution of problem (1.1) on . Let y(t) be the solution of the following ordinary differential equation
where
Since and , we have c_{1 }> 0. On the other hand, by virtue of (5.1), it is easy to see that δ_{1 }> δ_{2}. Then, it is obvious that there exists a constant 0 < T' < +∞ such that
Construct
where φ, ψ satisfying (1.7) and (1.8), respectively. Moreover, by the assumptions on initial data, we can take small enough constant y_{0 }such that
Now, we begin to verify that (
 u
 v
Similarly, we also have
On the other hand, ∀t ∈ [0, T], we have
and
Combining now (5.2)(5.6), we see that (
 u
 v
 u
 v
 u
 v
(ii) For p  1 > r, q  1 > s, we know mn = (p  1)(q  1). Under the assumption , we can choose A, B such that
Then, is a global supersolution of (1.1).
Since mn = (p  1)(q  1), we can choose constants l_{1}, l_{2 }> 1 such that
Next, we consider the following ordinary differential equation
where
Since , , we have c_{1 }> 0. On the other hand, in light of (5.7), it is easy to show that δ_{1 }> δ_{2}. Then, it is clear that y(t) will become infinite in a finite time T' < +∞.
Let
where φ(x), ψ(x) satisfies (1.7) and (1.8), respectively. Similar to the arguments for the case r > p  1, s > q  1, we can prove that (
 u
 v
(iii) For p  1> r, s > q  1, we know mn = s(p  1). Since , we can choose A, B such that
We can check is a global supersolution of (1.1).
Thanks to mn = s(p  1), we can choose constants l_{1}, l_{2 }> 1 such that
Let
where φ(x), ψ(x) are defined in (1.7) and (1.8), respectively, and y(t) satisfies the following Cauchy problem
where
Then, the left arguments are the same as those for the case r > p  1, s > q  1, so we omit them.
(iv) The proof of this case is parallel to (iii). The proof of Theorem 1.3 is complete. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors are very grateful to the anonymous referees and the editor for their careful reading and useful suggestions, which greatly improved the presentation of the paper. Dengming Liu is supported by the Fundamental Research Funds for the Central Universities (Project No. CDJXS 11 10 00 19). Chunlai Mu is supported in part by NSF of China (Project No. 10771226) and in part by Natural Science Foundation Project of CQ CSTC (Project No. 2007BB0124).
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