Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source
This paper is concerned with a nonlocal nonlinear diffusion equation with Dirichlet boundary condition and a source , , , , , , and , , which is analogous to the local porous medium equation. First, we prove the existence and uniqueness of the solution as well as the validity of a comparison principle. Next, we discuss the blowup phenomena of the solution to this problem. Finally, we discuss the blowup rates and sets of the solution.
Since the long-range effects are taken into account, nonlocal diffusion equations of the form have been widely used to model the dispersal of a species (see [1–7] and references therein). In fact, as stated in , if is thought of as the density of a species at the point at time and is thought of as the probability distribution of jumping from location to location , then is the rate at which individuals are arriving to position from all other places and is the rate at which they are leaving location to travel to all other sites. It is known that (1) shares many properties with the classical heat equation , such that bounded stationary solutions are constant, a maximum principle holds for both of them, and perturbations propagate with infinite speed (see ). However, there is no regularizing effect in general (see ).
Another classical equation that has been used to model diffusion is the well-known porous medium equation with . This equation also shares several properties with the heat equation, but there is a fundamental difference; in this case we have finite speed of propagation. Properties of solutions to the porous medium equation, particularly the blowup phenomena of the solution, have been largely studied over the past years. See, for example, [9–12] and references therein.
In [13, 14], a nonlocal model for diffusion that is analogous to the local porous medium equation is studied. In this model the probability distribution of jumping from location to location is given by when and 0 otherwise. In this case the rate at which individuals are arriving to position from all other places is , and the rate at which they are leaving location to travel to all other sites is . As before this consideration, in the absence of external sources, leads immediately to the fact that the density has to satisfy
In , Bogoya and Elorreaga studied the following nonlocal equation: They proved the existence and uniqueness of the solution as well as the validity of a comparison principle and also discussed the blowup phenomena of the solution for some sources.
In the present paper, we are concerned with the following nonlocal “Dirichlet” boundary value problem with a source: Here and . Let be a nonnegative, smooth function, with , supported in , symmetric, and strictly decreasing in . We assume that is a nonnegative function.
In this model, it is assumed that no individual can survive outside of the domain . Therefore, the density must be zero there. However, individuals are allowed to jump outside the domain (where they die instantaneously). This is what we call Dirichlet boundary conditions.
For the convenience of the statement of our results, denoting , some related definitions are introduced in the following.
Definition 1. A nonnegative function is a supersolution of the problem (4) if it satisfies The subsolution is defined similarly by reversing the inequalities. Furthermore, if is a supersolution as well as subsolution, then we call it a solution of the problem (4).
Definition 2. A solution of the problem (4) is called a global solution if supremum norm is finite for all .
Definition 3. If there is a time such that a solution of the problem (4) is bounded for all with , then blows up in finite time .
Now our main results can be stated as follows.
Theorem 4. If , , and , then there exists a unique solution to the problem (4).
Theorem 5 (blowup). Let be nonnegative and nontrivial. If , then the solution to the problem (4) blows up in finite time. If , the solution to the problem (4) is global. Moreover, if , one has the following estimate for the blowup time:
The rest of the paper is organized as follows. In Section 2, we prove the existence and uniqueness of the solutions for the problem (4) and show a comparison principle for the solution. In Section 3, we deal with the blowup phenomenon for the problem (4) by the method of supersolutions and subsolutions. That is, the estimate of the blowup time, the blowup rates, and sets of the solution of the problem (4) are discussed.
2. Existence and Uniqueness
This section is devoted to the proof of the existence and uniqueness of the solution to the problem (4) via Banach’s fixed point theorem. Simultaneously, the comparison principle for the solution of the problem (4) is also proved. To this end, it is convenient to give some preliminaries before giving its proof.
Fix and consider the Banach space with the norm We assume that a.e. in and . Let which is a closed subset of . We will obtain the solution of the problem (4) in the form , where is a fixed point of the operator defined by The following lemma is the main ingredient of the proof of Theorem 4.
Lemma 6. Let and be nonnegative functions such that and , and then Therefore, if is small enough, is a strict contraction in .
Proof. From the definition of , we have
Now set and . We have Since the integrands are nonnegative, applying Fubini’s theorem, we can get Therefore, we obtain
Since and , by the differential mid-value theorem, we deduce that
Furthermore, from the estimate (14) and (15), we get the desired estimate (10).
Next, we check that maps into . Since , for any , we have Hence, taking , we get that Thus, we conclude that .
Finally, choosing , we have
Therefore, if is small enough, is a strict contraction in . The proof is completed.
2.1. The Proof of Theorem 4
Proof. From Lemma 6, is a strict contraction in for small enough. By the Banach fixed point theorem, there exists only one fixed point of in . This proves the existence and uniqueness of the solution of (4) in the time interval . To continue, we may take as initial data and obtain a unique solution of (4) in the time interval . If , arguing as before with as the initial datum, it is possible to extend the solution up to some interval for certain . Hence, we can conclude that if the maximal time of the existence of the solution, , is finite, then the solution blows up in ; that is, Otherwise, the solution of the problem (4) is global.
Remark 8. If , then the solution of (4) for all .
Lemma 9. Let and be continuous supersolution and subsolution of the problem (4), respectively, and then for all .
Proof. By an approximation procedure we restrict ourselves to consider strict inequalities for the supersolution. Indeed, we can take () as a strict supersolution and take limit as at the end.
Hence, we consider strict inequalities for the supersolution and subsolution. Let for all . Arguing by contradiction, we assume that there exist a first time and some point such that and for all . Then, it holds that and we reach the desired contradiction.
3. Blowup Time, Blowup Rates, and Sets
3.1. The Proof of Theorem 5
Proof. For , integrating in and in (4), we get Applying Fubini’s theorem, we can obtain Let ; then , and we have That is, Using Hölder inequality, we have Since , we have that cannot be global; thus cannot be global either. Note that, by Theorem 4, in this case, we have that blows up in finite time in norm. Moreover, let , and we obtain Integrating the above inequality, we have Hence, it holds that where . Therefore, we can obtain the following estimate for the blowup time: For , let us consider the ODE problem Then, it follows that and for . We observe that . Therefore, is a global supersolution of the problem (4). Thus, is global by comparison.
Concerning the blowup rate, that is, the speed at which solutions are blowing up, we find the following result.
Theorem 10 (blowup rates). Let and be a continuous solution to the problem (4) which blows up at time . Then
Proof. For every , let be such that . Since for any , we have Changing the variable , then . Thus, we have Integrating the above inequality from and taking into account , we obtain To get the upper estimate, for any , we observe that In particular Taking into account (34) in this expression, we get Integrating (37) in , we obtain Taking the limit as , we will get the results.
The blowup set, that is, the set of points at which the solutions blow up, is defined as follows: Finally, we give the result concerning the blowup sets for the solution to the problem (4).
Theorem 11 (blowup sets). Let us consider the problem (4) with . Given and , there exists an initial condition, , such that the blowup set .
Proof. Given and we want to construct an initial condition such that
To this end, we will consider concentrated near and small enough away from .
Let be a nonnegative smooth function such that Now, let We want to choose large and small in such a way that (40) holds.
First, note that, thanks to the estimate (6), and taking large enough we can assume that is as small as we need. Now, using the upper bound for the blowup rate, we can obtain for any , where is small enough. Therefore, is a subsolution to And hence, if , we have
Now we just have to prove that a solution to (46) beginning with remains bounded up to , provided that and are small enough. To see this we use
So we have
Note that for and small it holds that . Indeed, we need For , we have , . If we choose the initial of the solution large enough to make sure large enough, then is small enough. So, for any fixed small , it holds that . So the (50) holds. From this, it is easy to prove that for all . Therefore, as . Going back to the equation verified by , for any , we have Integrating the above inequality from and using that , we have In terms of this bound implies that , for . From the boundedness of and (47), we get for every , as we desired.