Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms

Abstract

This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.

1. Introduction

In this paper, we consider the following problem: where ;

or (a ball in with radius ).

Moreover, for and satisfy the zero Dirichlet boundary conditions: are both nonzero, satisfying the monotonicity conditions:Moreover, in case of should satisfy compatibility conditions:As an application to system (1), a single equation of this system can be considered a simple model in population dynamics, [1, 2]:

where .

Let the domain represent a territory where a biological species live on. refers to the spatial density of individuals located near a point at time

In fact, the evolution of this density is the result of three types of mechanisms: displacement, birth, and death. For more details of deriving the evolution equation satisfied by , see [1].

Basically, under different assumptions on the mechanisms of accidental death, the corresponding term should more generally be a nondecreasing function of the density and its gradient .

Moreover, homogeneous Dirichlet’s conditions can be added to this model which, for instance, correspond to a nonviable environment in the boundary zone.

It is expected that, with a large size initial function (initial distribution of population ), the density becomes unbounded in a finite time . Therefore, Chipot and Weissler [3] studied the effect of the damping term in this equation on global existence or nonexistence.

The blow-up phenomena in Reaction-Diffusion equations have been intensively studied; see, for instance, [4–7]. One of the studied cases is the Cauchy problem of the semilinear heat equation: The second studied case is zero Dirichlet problem of the semilinear heat equation: where

For both cases (7) and (8), it has been proved in [8, 9] that if the initial function is nonnegative and suitably large, then blow-up occurs in a finite time. In [5, 10], it has been shown that the upper blow-up rate estimate for this equation is as follows: The blow-up properties of semiliear heat equations with negative sign gradient terms (damping terms) have been studied by some authors as in [3, 7, 11].

One of these equations is population model (6). For , it is well known that blow-up can only occur if ; see [3, 11, 12].

Moreover, if , then blow-up occurs at the center of , and this follows from the upper point-wise estimate: where , for ,

while , for

It is clear that , where

Therefore, the profile of blow-up solutions of (6) is similar to that of problem (8), where (see [12]), while if , the gradient term causes more effect on the plow-up profile and it becomes more singular.

Moreover, it has been proved in [4, 13, 14], that there are positive constants and , such that the upper and lower blow-up rate estimates for this equation, where , take the following form: In [15–18], the coupled system of Reaction-Diffusion equations was considered: where or

It was shown that if the initial functions satisfy , both being nonzero and large enough, then blow-up occurs in a finite time.

For the Cauchy problem associated with (12), it was proved in [16] that if then blow-up occurs in a finite time, where Later, in [4, 15], it was proved that the upper blow-up rate estimates of this system are as follows: for some

The system (1) has been studied in [19], where and is a bounded convex domain and It has been shown that if a classical solution of this system blows ups (becomes unbounded) in the W-norm, where then blow-up time for this problem can be estimated from below as follows: where

and is a constant which depends on the data.

For the blow-up times and applications of other parabolic systems with damping terms (such as Keller-Segel system with Neumann and Robin boundary conditions), we refer to [20, 21].

In this paper, with some restricted conditions on system (1), we show that the upper blow-up rate estimates for this solution and its gradients terms take the following forms:where are given in (15).

2. Local Existence and Blow-up

Set Since the system (1) is uniformly parabolic and its equations have the same principle parts and , also the growths of the nonlinearities in and with respect to the gradient terms are subquadratic; , and satisfying (5), it follows that the local existence and uniqueness of classical solution to the for system (1), where , with zero Dirichlet boundary conditions, are guaranteed by standard parabolic theory (see Theorem 7.1, [22, 23]).

i.e., there exists , such that Also, the gradient terms are bounded as long as the components of the solution are bounded; see [23].

In case of , these results can also be extended to the Cauchy problem associated with system (1) (see Theorem 8.1, [22, 24]).

Moreover, from the monotonicity assumptions (3) and (4) and since are nonnegative, it follows by the maximum principle [6] that in the interval of existence the solutions of system (1) are nondecreasing in time and nonnegative.

i.e., .

On the other hand, since the existence and uniqueness of system (1) can only be locally guaranteed and according to known blow-up results to the single equation (6), blow-up may occur in this problem in a finite time. Therefore, some authors were interested in studying the blow-up properties and numerical solutions of system (1); see for instance [19, 25].

3. Upper Blow-Up Rate Estimates

In the next theorem, we derive the upper blow-up rate estimates for any blow-up solution of system (1) and its gradients.

Theorem 1. Assume that , and satisfy the following two conditions: (i),(ii), where are given in (15).
Let be a blow-up solution of the Cauchy (Dirichlet) problem of system (1), with the above conditions, which blows up at There exist two positive constants and such that upper blow-up rate estimates for and are as follows:in .

Proof. For , set Clearly, each of is continuous, nondecreasing, and nonnegative function on Moreover, or as and that follows from blowing up at
It will be shown later that we can find such that for
So that consequently both and diverge as
In order to prove this theorem, we will use a rescaling method as in [4] and the proof will have five steps.
Step 1 (rescaling). If diverges as , then we can apply the following procedure.
Letting , we can choose such that Define the new rescaled functions as follows: where is a scaling factor and It is clear thatNext, we aim to show that is a solution of the system: where From assumption (ii), we get
Clearly,From (1) and (34), we get Hence, the first equation of the system (32) can be obtained by multiplying the last equation by The same way can be used to show that satisfies the second equation of the system (32).
Now, we restrict to to show that for
From (34), we obtain Clearly, From (32), (37), and (38), we get (36).
Moreover, for
On the other hand, from (26), we obtain If as , the same procedure can be repeated by changing the roles of and.
Step 2 (Schauder’s estimates). In this step, we find the interior Schauder’s estimates of the functions on the sets where Assuming that and satisfy in the conditionOur claim is as follows: for any positive and small enough values of , and , there exists a constant such that From (42), we deduce that , and are uniformly bounded functions in So, the functions , and are uniformly bounded in Therefore, the right hand side of each equation in (32) is uniformly bounded function in By applying the interior regularity theory (see [23]), we get locally uniform estimates in -norms. Consequently, on the right hand side of each equation in (32), we can obtain locally uniform estimates in Hölder norms . Therefore, the parabolic interior Schauder’s estimates (43) are held; see [23].
Step 3 (the proof of (25)). Suppose that the lower bound of (25) is not held. So, there is a sequence , such that as , and Thus, as .
Now, for each which plays the same role of , as in Step 1, we can scale about the corresponding point for each, where We get the corresponding rescaled solution : where is the scaling factor.
It is clear that satisfies, as in Step 1, the following problem: with for , where Clearly, From (44) and (48), we see that Thus and are bounded in for all
By applying Step 2, there is independent of , such that the uniform Schauder’s estimates of are as follows: Since is defined on a compact set, by the Arzela-Ascoli theorem, there exists a convergent subsequence, and it is denoted by
Since and are bounded, the limit point is a solution of the following system: Since , it follows that
Consequently, from the second equation of (53), we get Thus, which leads to a contradiction with (48), so the lower bound is proved.
If we change the roles of and , the upper bound of (25) can be proved similarly as in the last proof.
Step 4 (estimates on doubling ). Since as , is a continuous function. For any , the point can be defined as follows: Clearly, Take
We claim that there is which is independent of such that By supposing that this claim is not true, there is a sequence , as such that where For each , where , we can choose
As in Step 3, we scale about the corresponding point , and we can get the corresponding rescaled functions with the scaling factor: , which satisfies (47) with the following conditions:From (61) and (62), it follows that From (25), we conclude thatTherefore, (64) becomes By applying Step 2, we use the Schauder estimates for , and we can get a convergent subsequence in to the solution of system (53) in
Thus, we get a contradiction because under the assumption (i), all nontrival solutions of system (53) blow up in a finite time; see [17].
So, there is such that Step 5 (rate estimates).
For any , as in Step 4,
define , where
By (67), we have We can get by using as a new value of such that Thus, So, for any , we have where the sequence as
By adding the above inequalities, it follows thatThus, .
Using (25) results in Thus, From above, there exist positive two constants and such thatFrom the last two inequalities and the definitions of , it follows that there are such that Or, we can split the last estimates as follows: where

4. Conclusions and Future Studies

From Theorem 1 and its proof, we can point out the following conclusions:(i)By (25), it follows that the blow-up in system (1) can only occur simultaneously.(ii)The gradient terms are bounded for any .(iii)The upper blow-up rate estimates for both of systems (1) and (12) take the same forms. This means that, with the two conditions of Theorem 1, the gradient terms in the system (1) do not effect or make any changes on the profile of blow-up solutions.

Next, we point out some possible future research directions:(1)One may try to derive the blow-up rate estimates for problem (1), in case one or both assumptions (i) and (ii) of Theorem 1 are not satisfied.(2)For the semilinear system (12) defined in a ball, and under some restricted assumptions on (nonnegative and radial decreasing functions), it is well known that the blow-up can only occur at the center point (see [18]). However, it is unknown whether and under which condition this result can be extended to the system (1).

Data Availability

https://www.dropbox.com/home?preview=Data+Availability+Statement.pdf.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank Mustansiriyah University (http://www.uomustansiriyah.edu.iq) Baghdad-Iraq for its support in the present work.