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Singularity analysis for a semilinear integro-differential equation with nonlinear memory boundary
Journal of Inequalities and Applications volume 2014, Article number: 472 (2014)
In this paper, we study the blow-up singularity of a semilinear parabolic equation with nonlinear memory both in the reaction term and the boundary condition. We firstly establish the local solvability for a large class of semilinear parabolic equations with various nonlocal reaction terms. Secondly, we give a complete classification for the existence of a blow-up solution and a global solution. Next, we show that under some hypotheses the blow-up can only occur on the boundary of the domain.
In this paper, we devote our attention to the singularity analysis for the semilinear equation with nonlinear memory both in the reaction term and the boundary,
where , (), Ω is a bounded domain in having piecewise smooth boundary ∂ Ω with outward pointing unit normal ν. The initial data is a nontrivial, nonnegative, and continuous function on .
Models involving memory terms in reaction have arisen in studies of nuclear reactor dynamics [1, 2] and population dynamics , specifically in the case of logistic growth models involving both nondelayed and hereditary effects [4, 5]. Solvability, stabilization, and blow-up in finite time of solutions for a variety of generalizations of such models have subsequently been investigated in a number of previous works, e.g., [6–11]. Particularly, the blow-up properties of semilinear parabolic equation involving memory terms in a reaction,
coupled with a zero Dirichlet boundary condition has been completely studied (see [7–10]). Among other things, the authors obtained the following result: (1) Assume . If , then u blows up in finite time for sufficiently large , and u exists globally for sufficiently small . If then u blows up in finite time for any nonnegative nontrivial . (2) If , then u exists globally for any nonnegative . Meanwhile, the authors obtained the blow-up rate in the case of , in . Furthermore, some authors extended the above works for the semilinear case (1.2) to degenerate reaction-diffusion equations involving a nonlinear memory term and obtained a corresponding blow-up analysis (see, for example, [12–15]).
Memory terms in diffusion have been studied as well, arising in models of viscoelastic forces in non-Newtonian fluids [16, 17] and resulting from a modified Fourier law applied to anisotropic, nonhomogeneous media .
Despite the volume of work done on models incorporating memory in reaction, diffusion, or both, there appear to be very few appearances in the literature of diffusion models in which such terms are present in the boundary flux.
Recently, Deng et al. did some good work on the models with flux at the boundary governed by a nonlinear memory law in [19, 20]. Particularly, the authors studied the following model that has been formulated for capillary growth in solid tumors as initiated by angiogenic growth factors in :
Their primary result is that if , then every solution of (1.3) is global. On the other hand, if , then all nonnegative, nontrivial solutions blow up in finite time. Besides this, the authors proved that if , or , , blow-up can occur only on the boundary.
Motivated by above works, we investigate the blow-up properties of problem (1.1) in this paper. According to the aforementioned works, one may expect that the blow-up result of (1.1) is a combination of (1.2) and (1.3) to some extent. In fact, we shall prove that if and , solution exists globally for any nonnegative . We also find that if or , a blow-up singularity occurs and all nonnegative nontrivial solutions blow up in finite time. We notice that the main idea as regards the time-integral nonlocal problems is that only when the time is large, the time-integral term dominates the evolution of the solutions. Therefore, for problem (1.2), a solution still maybe exists globally even when . For our problem (1.1), however, if or , there is no global solution. Thus, one can see that the nonlinear memory boundary plays an important role in accelerating the occurrence of a blow-up singularity.
The remaining part of this paper is organized as follows. In Section 2, we prove the local solvability of a wide class of integro-differential equations of the parabolic type involving nonlinear memory terms, which include (1.1). In Section 3, we give the comparison principle which will be used later for nonnegative solutions to (1.1). In Section 4, we establish the global existence and finite time blow-up result. In the last section, we shall investigate the blow-up set. We will prove that the blow-up may only occur on the boundary of the domain in some cases.
2 Local solvability
In this section we derive the local solvability for a large class of semilinear parabolic equations with various nonlocal reaction terms and memory boundaries, which include (1.1). Furthermore, we give the local existence theorem for (1.1), where the nonlinearity is merely locally Hölder continuous.
Theorem 2.1 Assume that F, f, H, and h are all locally Lipschitz continuous functions. Consider the following problem:
For every , then there exists a such that the problem has a unique classical solution .
Proof Set , where .
Given , define
Here is the Green’s function for the heat equation with homogeneous Neumann boundary condition. (We refer the readers to [24, 25] and  for its construction and properties.) As F, f, H, and h are all locally Lipschitz continuous functions, we may assume that, for all , there exists , , , such that, for all with , the functions F, f satisfy
and H, h satisfy
Noticing that and for constants with , we have
which implies if T is small enough.
So, Φ is a mapping from Σ into itself.
Similarly, we obtain
Therefore, if is small enough, then we see that Φ is a strict contraction. Thus, Φ has a unique fixed point by Banach’s fixed point theorem. This implies that, for any , there exists a unique local solution in the integral sense for small enough.
Concerning the regularity, we can see that the corresponding solution u is automatically in from the standard bootstrap argument. On the other hand, the continuity of the solution follows from (1.1) itself (see  for details). □
Of course, when in problem (1.1), the above local well-posedness does not apply to (1.1). However, for problem (1.1), we still have the following local existence theorem.
Theorem 2.2 For every nonnegative nontrivial , there exists a such that problem (1.1) has a unique nonnegative classical solution .
Proof We may only give the proof for the case that (). Let
Note that , () are monotone decreasing with respect to n and
For any fixed n, , () are non-decreasing, locally Lipschitz functions of z.
Let be a sequence of solutions such that
Then we have a unique classical approximated solution by Theorem 2.1. Since , by the maximum principle we know that , and by the comparison theorem (see Lemma 3.3) we see that is monotone decreasing. Hence, there exists a bounded nonnegative function , which corresponds to the continuous solution of (1.1). On the other hand, we get the additional regularity of u from the standard argument. When is nontrivial, the uniqueness follows from the strong maximum principle. □
3 Comparison principle
In order to use the super-sub-solution technique, we next introduce the definition of the super- and the sub-solution and the comparison principle for problem (1.1).
Definition 3.1 A function is called the super-solution of problem (1.1) if and satisfies
Similarly, we can obtain the definition of sub-solution of problem (1.1) by all inequalities revised.
Lemma 3.1 Suppose that and satisfies
where () are bounded functions and (). Then .
Proof Let be a positive smooth function and satisfies , is a constant to be determined later. Let , where is a constant to be determined. Then, if we choose , , then
Suppose that attains negative minimum at . If , then , , at . On the other hand, we know from the first inequality of (3.3) that
This is a contradiction.
If , then at ,
which contradict to . Therefore, for any , we have . The same for . □
Lemma 3.2 Suppose that , (). If and are the nonnegative super-solution and sub-solution of (1.1), respectively, then in .
Proof Let . It is easy to verify that w satisfies (3.2), where () are such that
From Lemma 3.1, we know . □
Remark 3.1 From the above proof, we know when or () and there exists some such that , , the functions () are still bounded. Therefore the conclusion of the lemma is valid in this case.
Using Lemma 3.1, we could obtain another version of the comparison theorem, which is useful in the proof of the local existence of the solution.
Lemma 3.3 Let , () be non-decreasing locally Lipschitz functions. Suppose that such that
Then in .
Proof The proof is similar to that of Theorem 3.3 in  by using Lemma 3.1. We only give a sketch of the proof.
We only need to prove in . Set . Let () be continuous functions defined by
Then satisfies the condition of the lemma. Thus, in . □
4 Global existence and finite time blow-up
In this section, we shall determine when the solution of problem (1.1) exists globally or blows up in finite time. We first establish the global existence result.
Theorem 4.1 If and , then the solution of problem (1.1) exists globally for any nonnegative .
Proof We shall construct a super-solution of (1.1). From , we know that there exists a function satisfying
Denote and . Define
Direct calculations show that
We notice that
Therefore, is a super-solution of problem (1.1). From the comparison principle, we know the solution of (1.1) exists globally. □
In the remainder of this section, we shall establish the finite time blow-up result of problem (1.1). We have the following theorem.
Theorem 4.2 If or , then all nonnegative solutions of (1.1) blow up in finite time.
To prove this theorem, we need to consider first the following problem:
Lemma 4.1 For any positive constant c, if , then all nonnegative solutions of (4.1) blow up in finite time.
Proof From now on, we use () to denote various positive constants. Let be the Green’s function for the heat equation with homogeneous Neumann boundary condition. Then we have the following representation formulas for the solution of (4.1):
As is well known, the Green’s function satisfies (see, e.g., )
By (4.2), (4.3), and Jensen’s inequality, we have
Then it follows from (4.4) that
Integrating this inequality from 0 to t, we obtain
Assume to the contrary that (4.1) has a global solution u. Then, for any , we have
Thus, for any , where
Multiplying the first equality in (4.5) by and integrating over , we obtain
Integrating this equality again from T to 2T, one can get
This leads to a contradiction when T is large enough. Therefore, problem (4.1) has no global solution. □
Remark 4.1 We can also easily reach our conclusion by using the comparison principle. In fact, the solution of the corresponding Dirichlet problem,
is the sub-solution of (4.1). We know from Theorem 5.1 of  and Theorem 3.3 of  that if the solution of this first initial boundary value problem blows up in finite time for any nonnegative nontrivial initial data. However, applying the representation formula of the Neumann problem (4.1), we gave a completely different approach here.
Now we are ready to prove Theorem 4.2.
Proof of Theorem 4.2 When , we first consider the following problem,
If , we know (, ) by the maximum principle. Let
where , then z satisfies
So z is a super-solution of problem (4.1). As , Lemma 4.1 shows that z blows up in finite time, so does ϕ.
Now we consider the case of . If , the solution of the following ODE:
is the sub-solution of problem (4.7). The fact that shows that , which leads to the finite time blow-up of this ODE. So does the solution of (4.7).
If and , for any there exists a constant such that the solution to (4.7) satisfies
Here we used the fact that the solution of the heat equation with homogeneous Neumann boundary condition is a sub-solution of problem (4.7).
Let , where and , then we have
Next we consider the following problem:
Proceeding analogously to the proof of Lemma 4.1 and the case , we can show that blows up in finite time. Since is a super-solution of (4.7), ϕ cannot exist globally. The solution to problem (1.1) is a super-solution of (4.7).
Therefore, when , all solutions of (1.1) blow up in finite time.
When , we know from  that solutions of
blow up in finite time. The comparison principle tells that all solutions of (1.1) blow up in finite time. □
5 Blow-up set
Examples in  indicate that the blow-up may occur in the interior of the domain for some semilinear parabolic equation if the heat supply through the boundary is fast enough. However, we shall prove in this section that the blow-up of problem (1.1) will occur only at the boundary of the domain in some cases. This implies that the diffusion term is the dominating term in the interior of the domain for these cases. To this end, we need a lemma as follows. We will prove this lemma by the idea introduced in . However, we need some careful modification due to the appearance of the nonlinear memory reaction term.
Lemma 5.1 Suppose that the function is continuous on the domain and satisfies
for some (). Then, for any ,
Proof We will prove this lemma in a similar way to Theorem 4.1 of .
where . By approximating the domain from inside if necessary, we may assume without loss of generality that ∂ Ω is smooth, say . Henceforth, the function is in if ε is small enough. As stated in Theorem 4.1 of , we could extend to a function on such that , on and
for some small enough and .
Denote , then there exist , such that for any (). If , then direct calculation shows that
Choose , then for any . Moreover, as , we can choose to be large enough so that and for . Then the comparison principle implies that , and
where , .
By a similar way used in the case of ; we could choose suitable l and such that for any . Moreover, and for .
Therefore, when , we get
from the comparison principle. □
Now we are ready to give our main result as regards the blow-up set. To this end, we need the following hypothesis.
(H1) , i.e. for all .
Theorem 5.1 If and one of the following cases occurs, the blow-up of problem (1.1) occurs only at the domain of the boundary:
, , and (H1) is valid,
Proof is the Green’s function for the heat equation with a homogeneous Neumann boundary condition. Then we have the following representation formulas for the solution of problem (1.1):
Next, we consider three cases, respectively.
Case 1 (, ). Set
Therefore, (5.1) and Jensen’s inequality show that
Multiplying both sides of the above inequality by and integrating over , we obtain
Integrating this inequality over , one can get
We now take an arbitrary with . For this , we then take such that , and . It is well known that, for any ,
where is a positive constant depending on ε. Then by (5.1), (5.4),
As , Lemma 5.1 implies that the blow-up occurs only at the boundary.
In the remaining two cases, we set
where . In view of Case 1 and Lemma 5.1, it suffices to prove that satisfies a similar estimate to (5.3). Precisely, if we can prove that
for some positive constants , β, then the blow-up can only occur on the boundary ∂ Ω.
Case 2 (). and show that . From (4.3), (5.1), and Jensen’s inequality, we can obtain that
Integrating this inequality over , we have
Case 3 (). As , we know from  that, for some , there exists such that, for , ,
Denote , then
Using (5.1), (5.2), (5.6), and Jensen’s inequality, one can easily get
From this we can obtain
Integrating over , we know that
Since , we can conclude that blows up only on the boundary in a similar way to Case 1 by Lemma 5.1. □
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The authors are very grateful to the referees for their valuable comments, which greatly improved the manuscript. This work is supported by Scientific Research Found of Sichuan Provincial Education Department (14ZA0119), Xihua University Young Scholars Training Program, and Applied Basic Research Project of Sichuan province (2013JY0178).
The authors declare that they have no competing interests.
YW carried out most of the studies and wrote the manuscript. JC did some work in Section 3 to Section 5. CH participated in the study of blow-up set. All authors read and approved the final manuscript.