Blowup for Nonlocal Nonlinear Diffusion Equations with Dirichlet Condition and a Source

and Applied Analysis 3 The following lemma is the main ingredient of the proof of Theorem 4. Lemma 6. Let w 0 and z 0 be nonnegative functions such that w 0 , z 0 ∈ L 1 (R) and w, z ∈ X t0 , and then 󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 F w0 (w) (x, t) − F z0 (z) (x, t) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ [1 + p(A + c) p−1 ] (1 − e −t0 ) |‖w − z‖| + 󵄩 󵄩 󵄩 󵄩 w 0 − z 0 󵄩 󵄩 󵄩 󵄩L 1 (Ω) . (10) Therefore, if t 0 is small enough,F w0 is a strict contraction inX t0 . Proof. From the definition of F w0 , we have ∫ R 󵄨 󵄨 󵄨 󵄨 󵄨 F w0 (w) (x, t) − F z0 (z) (x, t) 󵄨 󵄨 󵄨 󵄨 󵄨 dx = ∫ R 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫ t 0 e −(t−s) [∫ R (J( x − y


Introduction
have been widely used to model the dispersal of a species (see [1][2][3][4][5][6][7] and references therein).In fact, as stated in [7], 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 ∫ R  ( − )(, ) is the rate at which individuals are arriving to position  from all other places and −(, ) = − ∫ R  ( − )(, ) 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   = Δ, such that bounded stationary solutions are constant, a maximum principle holds for both of them, and perturbations propagate with infinite speed (see [7]).However, there is no regularizing effect in general (see [8]).
Another classical equation that has been used to model diffusion is the well-known porous medium equation   = Δ  with  > 1.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][10][11][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 (( − )/(, ))(1/(, )) when (, ) > 0 and 0 otherwise.In this case the rate at which individuals are arriving to position  from all other places is ∫ R (( − )/(, )), and the rate at which they are leaving location  to travel to all other sites is −(, ) = − ∫ R (( − )/(, )).As before this consideration, in the absence of external sources, leads immediately to the fact that the density (, ) has to satisfy In [15], 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  ≥ 1 and  ≥ 0. Let  : R → R be a nonnegative, smooth function, with ∫ R () = 1, supported in [−1, 1], symmetric, and strictly decreasing in [0, 1].We assume that  0 ∈  1 (R) 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 Ω = (−, ), some related definitions are introduced in the following.
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).
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.

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  0 > 0 and consider the Banach space   0 := ([0,  0 ];  1 (R)) with the norm We assume that 0 ≤  0 () ≤  a.e. in Ω and  = 2 + 1.Let which is a closed subset of   0 .We will obtain the solution of the problem (4) in the form (, ) = (, ) + , where  is a fixed point of the operator   0 :   0 →   0 defined by The following lemma is the main ingredient of the proof of Theorem 4.
Proof.From the definition of   0 , we have   Furthermore, from the estimate ( 14) and (15), we get the desired estimate (10).

The Proof of Theorem 4
Proof.From Lemma 6,   0 is a strict contraction in   0 for  0 small enough.By the Banach fixed point theorem, there exists only one fixed point of   0 in   0 .This proves the existence and uniqueness of the solution of (4) in the time interval [0,  0 ].To continue, we may take (,  0 ) as initial data and obtain a unique solution of (4) in the time interval [0,  1 ].
< ∞, arguing as before with (⋅,  1 ) as the initial datum, it is possible to extend the solution up to some interval [0,  2 ) for certain  2 >  1 .Hence, we can conclude that if the maximal time of the existence of the solution, , is finite, then the solution blows up in  1 (Ω) norm; that is, Otherwise, the solution of the problem (4) is global.
To complete the proof of Theorem 5, we introduce the comparison principle for the problem (4) which is a very useful tool in studying diffusion problems.Lemma 9. Let  and  be continuous supersolution and subsolution of the problem (4), respectively, and then (, ) ≤ (, ) for all (, ) ∈ Ω × [0, ).
Proof.By an approximation procedure we restrict ourselves to consider strict inequalities for the supersolution.Indeed, we can take (, ) +  +  ( > 0) as a strict supersolution and take limit as  → 0 at the end.

Blowup Time, Blowup Rates, and Sets
Once the existence and uniqueness of the solutions to the problem ( 4) are proved, we begin to analyze the blowup phenomenon for the problem (4).
Concerning the blowup rate, that is, the speed at which solutions are blowing up, we find the following result.
Taking the limit as  → , we will get the results. 6 Abstract and Applied Analysis The blowup set, that is, the set of points at which the solutions blow up, is defined as follows:  () = { ∈ Ω; there exists a finite time  with  (, ) → ∞ as  ↗ } . (39) Finally, we give the result concerning the blowup sets for the solution to the problem (4).
Proof.Given  0 ∈ Ω and  > 0 we want to construct an initial condition  0 such that To this end, we will consider  0 concentrated near  0 and small enough away from  0 .