Singularity analysis for a semilinear integro-differential equation with nonlinear memory boundary
© Wang et al.; licensee Springer. 2014
Received: 4 April 2014
Accepted: 10 November 2014
Published: 26 November 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.
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 .
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.
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.
For every , then there exists a such that the problem has a unique classical solution .
Proof Set , where .
which implies if T is small enough.
So, Φ is a mapping from Σ into itself.
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 .
For any fixed n, , () are non-decreasing, locally Lipschitz functions of z.
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).
Similarly, we can obtain the definition of sub-solution of problem (1.1) by all inequalities revised.
where () are bounded functions and (). Then .
This is a contradiction.
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 .
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.
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.
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 .
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.
Lemma 4.1 For any positive constant c, if , then all nonnegative solutions of (4.1) blow up in finite time.
This leads to a contradiction when T is large enough. Therefore, problem (4.1) has no global solution. □
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.
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.
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).
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).
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.
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.
Proof We will prove this lemma in a similar way to Theorem 4.1 of .
for some small enough and .
where , .
By a similar way used in the case of ; we could choose suitable l and such that for any . Moreover, and for .
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 .
, , and (H1) is valid,
Next, we consider three cases, respectively.
As , Lemma 5.1 implies that the blow-up occurs only at the boundary.
for some positive constants , β, then the blow-up can only occur on the boundary ∂ Ω.
Since , we can conclude that blows up only on the boundary in a similar way to Case 1 by Lemma 5.1. □
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).
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