Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption
We investigate a nonlocal reaction diffusion equation with absorption under Neumann boundary. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of solutions blowing up in finite time, or for the global existence and boundedness of all solutions. For the blowup solutions, we also study the blowup rate estimates and the localization of blowup set. Moreover, we show some numerical experiments which illustrate our results.
In this paper, we devote our attention to the singularity analysis of the following nonlocal diffusion equation: Here is a bounded connected and smooth domain, which contains the origin, and is a nonnegative, bounded, symmetric radially and strictly decreasing function with , and , , are all positive constants. We take the initial datum, , nonnegative and nontrivial.
Equation in (1.1) is called a nonlocal diffusion equation in the sense that the diffusion of the density at a point and time does not only depend on , but on all the values of in a neighborhood of through the convolution term. Maybe the simplest linear version of nonlocal model (1.1) is In recent years, the linear equation (1.2) and its variations have been widely used to model diffusion process, for example, in biology, dislocations dynamics, phase transition model, material science, and network model and so forth (see [1–7] and the references therein). The idea hidden inside those model is simple to understand. As stated in , if is thought of as a density at the point and time and is thought of as the probability distribution of jumping from location to location , then the convolution is the rate at which individuals are arriving at from all other places and is the rate at which they are leaving location to travel to all other sites.
In the past decades, some works have shown that (1.2) shares many properties with the classical heat equation such as the existence of constant bounded stationary solutions and the maximum principle, but there is no regularizing effect in general for the former . Therefore, as mentioned in [9–11], it is an interesting topic to compare the properties of solutions to the nonlocal diffusion equation with that of the corresponding local diffusion cases.
To motivate our works, we would like to remark that, in recent years, (1.3) and its variations have been extensively studied. In particular, a considerable effort has been devoted to studying the blowup properties of the following classical diffusion equation with reaction () and/or absorption () under Dirichlet or Neumann boundary which provides a simple biological or physical model. For instance, by constructing the self-similar weak subsolutions, Bedjaoui and Souplet  obtained the following conclusion for (1.4) with under Dirichlet or Neumann boundary: if , then all solutions are global. If , there exist solutions of (1.4) which blow up in finite time. In the critical case , the results may depend on the size of the coefficient . In the Dirichlet boundary case, Xiang et al.  also studied the blowup rate estimates and obtained the following results: if and the solution of (1.4) blows up at , then there exists constants such that
In this paper, we will deal with the blowup properties of the nonlocal diffusion problem (1.1) and compare them with that of problem (1.4). In our model (1.1), the absorption term represents a rate of consumption due to an internal reaction, and we are integrating in and thus imposing the condition that the diffusion takes place only in , which means that the individual may enter or leave the domain. This is so called Neumann boundary conditions, see [14, 15]. We remark that García-Melián and Quirós  recently proved the existence of a critical exponent of Fujita type for the nonlocal diffusion problem As mentioned in , nonlocal diffusion systems are more accurate than the classical diffusion systems in modeling the spatial diffusion of the individuals in some biology areas, such as embryological development process. For more study about the nonlocal diffusion operator, we refer to [8, 18–22] and references therein.
It is noteworthy that the method used in [12, 13] for problem (1.4) is invalid in our current setting due to the appearance of the nonlocal diffusion term. For example, instead of constructing self-similar weak subsolutions, we will use technique of integration to prove the finite time blowup. As far as the blowup rate is concerned, the scaling argument in  is not applicable.
We now state our main results. Our first result determines the complete classification of the parameters for which the solution blows up in finite time or exists globally.
(i) If , then all solutions of (1.1) are global. Moreover, if or and , all solutions are uniformly bounded, while if and , there exist unbounded global solutions.
(ii) If , then (1.1) with large initial data have solutions blowing up in finite time, while the solutions of (1.1) with small initial data exist globally.
Once we have obtained the values of the parameters for which blowup occurs, the next step is to concern the blowup rate, that is the speed at which solutions are blowing up. Different from the result of problem (1.4), we could have a unified upper and low estimate here.
Theorem 1.2. Let and be a solution of (1.1) blowing up at time . Then
Remark 1.3. From this result we find that the nonlocal diffusion term plays no role when determining the blowup rate and the blowup rate is just same as that of the ODE . And this phenomena is the same as that of local diffusion case.
Next we consider the spacial location of the blowup set. As usual, the blowup set of solution is defined as follows: where is the maximal existence time of . For a general domain we can localize the blowup set near any pint in just by taking an initial condition being very large near that point and not so large in the rest of the domain. This is the following result.
Theorem 1.4. Let . For any and , there exists an initial data such that the corresponding solution of (1.1) blows up at finite time and its blowup set is contained in .
Next we consider the radially symmetric case. In this case, single point blowup occurs.
Theorem 1.5. Let and . If the initial data is a radial nonnegative function with a unique maximum at the origin, that is, , for , and , then the blowup set of the solution of (1.1) consists only of the original point .
The remainder of this paper is organized as follows. In Section 2, we give the existence and uniqueness of the solutions as well as the comparison principle. In Section 3, we prove the blowup and global existence condition. And then we prove the blowup rate and blowup set results in Sections 4 and 5, respectively. And in the last section we will give some numerical experiments to demonstrate our results.
2. Existence, Uniqueness, and Comparison Principle
We begin our study of problem (1.1) with a result of existence and uniqueness of continuous solutions and comparison principle.
Firstly, existence and uniqueness of solutions is a consequence of Banach's fixed point theorem. We look for satisfying (1.1). Fix , and consider the Banach space with the norm
We define the following operator
Theorem 2.1. For every there exists a unique solution of (1.1) such that and (finite or infinite) is the maximal existence time of solution.
Next, we will study the comparison principle. As usual we first give the definition of supersolution and subsolution.
Definition 2.2. A function is a supersolution of (1.1) if it satisfies Subsolutions are defined similarly by reversing the inequalities.
To obtain the comparison principle for problem (1.1), we first give a maximum principle.
Lemma 2.3. Suppose that is nontrivial and satisfies with for , where is a bounded function, then for and .
Proof. We first show for . Assume that is negative somewhere. Let . If we take a point where attains its negative minimum, there holds and
which is a contradiction. Thus for . And so does .
Now, we suppose for some ; that is, attains its minimum at from the first step. Notice that the hypotheses on imply , so that implies that for in a neighborhood of . Thus a standard connectedness argument yields . This is a contradiction. So we obtain our conclusion.
Lemma 2.4. If , and , are super and subsolutions to (1.1), respectively then for every .
Remark 2.5. When or , the conclusion is also validity if and are bounded away from 0.
3. Blowup and Global Existence
In this section, we will analyze the blowup condition and give the proof of Theorem 1.1.
Proof of Theorem 1.1. (i) We only need to look for a global supersolution of (1.1). Indeed, it is easy to construct spacial homogeneous global supersolution of (1.1). To see this, we set , where and are positive constants to be determined.
For any given initial data , we note that for sufficiently large and is bounded away from 0. Thus by the comparison principle and Remark 2.5, to make a supersolution of (1.1), we only need to show the existence of and satisfying If , for any given , we can take such that (3.1) holds. If and thus , we can choose and satisfying , which make (3.1) validity.
Next, we show all global solutions are uniformly bounded when or and . In fact, (1.1) has constant supersolution whenever or and . To see this, we choose large enough such that which imply that is a supersolution of (1.1).
At last we show there exist global unbounded solutions when and . Indeed, let It is easy to see that if , is a subsolution of (1.1). It is obvious that when , is unbounded.
(ii) We first show that if the initial data is large enough, solutions of (1.1) blow up in finite time.
In the case of . Integrating (1.1) in and applying Fubini's theorem, we get Using Hölder's inequality, we could get where is assumed to be the measure of . Given positive constant and , we have by the comparison principle that the solution of problem (1.1) satisfies . Thus Then we use Jensen's inequality to obtain where is a positive constant independent of the solution . From this inequality, we could easily obtain that blow up in finite time.
In the case of , it follows from and Jensen's inequality that Substituting this inequality into (3.4), we obtain
Therefore, if we take the initial data large enough such that , then blows up in finite time. So does .
Next we show when the initial data is small, solutions of (1.1) exist globally. Consider constant . Let . Then . Henceforth, if , is a supersolution of (1.1). From the comparison principle, we know solutions of (1.1) are global in this case.
4. Blowup Rate Estimate
In this section, we study the blowup rate and prove Theorem 1.2.
Proof of Theorem 1.2. Let . It is easy to see that is Lipschitz continuous and thus it is differential almost everywhere . From the first equality of (1.1) we have
at any point of differentiability of . Here we used . Noticing that and integrating (4.1) from to , we obtain
Next we will establish the upper estimate. For any , we have In particular,
From the lower estimate (4.2) we get
Integrating in , we get combining with (4.2), the conclusion of Theorem 1.2 is proved if one takes the limit as .
5. Blowup Set
Next we will concern the blowup set for the solution to problem (1.1). We will first localize the blowup set near any point in just by taking an initial condition being very large near that point and not so large in the rest of the domain.
Proof of Theorem 1.4. Given and , we could construct an initial condition such that
In fact, we will consider concentrated near and small away from .
Let be a nonnegative smooth function such that and for .
Next, let We want to choose large and small such that (5.1) holds.
First we can assume that is as small as we need by taking large enough. Indeed, we have from the proof of Theorem 1.1.
Now, from the proof of blowup rate, we have Henceforth, for any , Therefore, if , then is a subsolution to which shows
Next, we only need to prove that a solution to (5.6) remains bounded up to , provided that and are small enough.
Let Then satisfies which show that for and small ( is small if is large), we have So for all . From this and Lemma 4.2 of , we know Combining the equation verified by we obtain that, for given positive constant , there exists such that for .
Let be a solution of with . Integrating this equation we get By a comparison argument we could get that for every , Now we go back to . We have then Integrating form to , one could get Using (5.15) and , we have as .
And thus from (5.18), we get
As , we have This implies that , for . From the boundedness of and (5.7) we get for every , as we wished.
Next, we will consider the radial symmetric case, that is, the proof of Theorem 1.5. For the convenience of writing, we only deal with the one dimensional case, . The radial case is analogous; we leave the details to the reader.
Proof of Theorem 1.5. Under the hypothesis on the initial condition imposed in Theorem 1.5 we have that the solution is symmetric and in from the stand parabolic theorem and Lemma 4.1 of . Therefore the solution has a unique maximum at the origin for every .
Let us perform the following change of variables Our remainder proof consist of two steps.
Step . We first prove the only blowup point that verifies the blowup estimate (1.7) is . And this shows that for , does not converge to as .
We conclude by contradiction. Assume that for a .
Let . Then where and are between and . Hence for some positive constant.
Integrating the above inequality, we obtain Remember that , , we have And this implies that implies that as for given . Hence for some .
Hence Using this fact, we have This contradiction proves our claim.
Step . We will show the only possible blowup point is .
Remembering the transform (5.22), satisfies Note that the blowup rate of implies that for every . Therefore, From this we know that if there exists such that , then as (see Lemma 4.2 in ).
Moreover, if there exists such that then blows up in finite time . This follows from Lemma 4.3 of  using that
Thus if does not converge to zero and does not blow up in finite time, then satisfies Henceforth, As is continuous, bounded and does not go to zero, we conclude that .
Now we could conclude that verifies , or , or blows up in finite time.
From Step 1 we know for , is bounded and does not converge to , so as . Combined with inequality (5.32), we could get for any .
By a comparison argument as in the proof of Theorem 1.4, it follows that Going back to the equation verified by we obtain Integrating we get On the other hand, (5.37) implies that as . Henceforth, Using that , one could have Remembering the transform (5.22), we have And so our proof is complete.
6. Numerical Experiments
At the end of this paper, we will use several numerical examples to demonstrate our results about the location of blowup points. For this purpose, we discretize the problem in the spacial variable to obtain an ODE system. Taking and , we consider the following system: Next we choose , , and
In Figure 1 we choose a nonsymmetric initial condition very large near the point , . We observe that the blowup set is localized in a neighborhood of .
Next we choose a symmetric initial condition with a unique maximum at the origin, . We observe that the solution blows up only at the origin, Figure 2.
Y. Wang is supported by the Key Scientific Research Foundation of Xihua University (no. Z0912611), the Scientific Research Found of Sichuan Provincial Education Department (no. 09ZB081) and the Research Fund of Key Disciplinary of Application Mathematics of Xihua University (no. XZD0910-09-1). Z. Xiang is supported by NNSF of China (11101068), the Sichuan Youth Science & Technology Foundation (2011JQ0003), the Fundamental Research Funds for the Central Universities (ZYGX2009X019) and the SRF for ROCS, SEM. J. Hu is supported by the Scientific Research Found of Sichuan Provincial Education Department (no. 11ZB009).
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