On Nonlinear Nonlocal Systems of Reaction Diffusion Equations

Abstract

The reaction diffusion system with anomalous diffusion and a balance law ,  , , , is con sidered. The existence of global solutions is proved in two situations: (i) a polynomial growth condition is imposed on the reaction term when ; (ii) no growth condition is imposed on the reaction term when .

1. Introduction

In this paper, we consider the system of nonlinear and nonlocal in space reaction diffusion equations supplemented with the initial conditions where the initial data are given positive bounded functions.

Here the nonlocal operator or accounts for anomalous diffusion (see, e.g., [1–3]) and can be defined via the Fourier transform pair and as where is the Schwartz class of smooth real rapidly decreasing functions, or equivalently (see [4]) by the formula with a normalizing constant, and denotes the usual norm of .

A typical type of system under our consideration is given by the irreversible molecular combination where and are two chemical species. If and represent the concentrations of the species and , respectively, then according to the law of mass action due to Gulberg and Waage, the reaction diffusion system describing the chemical reaction can be written as where . This system has been studied by Masuda [5] via a judicious Lyapunov functional, Hollis et al. [6] by using the duality argument, Collet and Xin [7] in the case of the Euclidean space.

Let us now dwell for a while on the available literature concerning anomalous diffusion equations. Fractional differential equations have been used as effective mathematical tools for modeling diffusive processes associated with subdiffusion (fractional in time), superdiffusion (fractional in space), or both. Further examples can be found in physics, mathematical biology, or hydrology. These equations also appear in finance because of the relationship with certain option pricing mechanisms and heavy tailed stochastic processes [8]. In water resources, fractional models have been used to describe chemical and contaminant transport in heterogeneous aquifers [9]. In spatial complex environment, reaction diffusion equation may not obey Fick’s Law [10]. One idea is to replace the flux, say , by its fractional counterpart [11]: where is the diffusion tensor and is the Riemann-Liouville fractional gradient, where with similar expressions for and [12]. The fractional Fick’s Law for (8) implies nonlocality in space and in time. This modification, in the absence of external force, leads to the fractional diffusion equation Equivalently, in the isotropic setting [13], the space fractional reaction diffusion can be written as where is the fractional Laplacian operator; see also the valuable contribution of Douglas [14] for the use of the fractional Laplacian in polymer sciences.

In our consideration, we take into account the diffusion of two interacting species, diffusing at different rates.

The reaction term is locally Lipschitz continuous, namely, for all .

Further, it is assumed that there exist positive numbers ,  , and such that for all with , and (Note that for all .)

We first prove that system (1)–(3) admits global solutions for reaction terms of polynomial growth relying on the duality argument that has been used by Hollis et al. [6] for the case when the space variable belongs to a bounded domain and . Notice that estimates obtained by this method have been recently improved by Cañizo et al. [15] in the same case . In case of , the duality method has been used successfully by Fitzgibbon et al. [16] still in the case .

A central role in the proof is played by a recent regularity result due to Zhang [17] for the solution of the backward heat equation supplemented with the condition which will be stated in Section 2.

Next, we prove our second result; namely, global solutions of problem (1)–(3) exist for any growth of the reaction terms whenever .

Our second result has to be compared in some sense with that of Martin and Pierre [18]. It has been shown in [18] that the following problem admits global solutions for any nonlinearity under the condition : supplemented with positive and bounded initial data.

The result of [18] is recalled in the appendix for the reader in order to compare our result with the result of Martin and Pierre.

The result of [18] has been extended by Kanel and Kirane [19] for the triangular system where is a bounded regular domain with boundary , is the outward normal derivative to , and are the positive diffusion constants.

2. Preliminary Results

Notation. Consider , , and .

The proof of our first result is based on a recent lemma of Zhang [17] (Lemma 2) and a known interpolation inequality (Lemma 3).

Lemma 1. Let be the linear semigroup generated by the following linear anomalous diffusion problem: Let and . Then the solution of (19) satisfies the estimate for and .

The proof of this lemma follows from the Young inequality combined with scaling properties of the kernel with where is the ordinary inner product at the points and .

The lemma is used for the local existence , as well as for the global existence .

Lemma 2. Let and suppose that . Then (15)-(16) has a unique positive solution such that . Moreover, there exists a constant , independent of such that

Lemma 3. Let be a Banach space and a positive operator on . Then, for , there exists a constant such that for (the domain of )

The proof of our second result is based on the following interesting lemma of Lopez-Mimbela and Morales [20].

Let be the continuous transition density of the symmetric stable process in of index , , which is uniquely determined by

Lemma 4. Let , be the transition density of the symmetric -stable process in , . If , then there exists a constant such that, for every and , If in addition , then

As the proof is nice and instructive, we present it for the convenience of the reader.

Proof. By Theorem  2.1 [21], we have If , then as . Hence, there exists a constant such that for all . Since is continuous and is compact, there exists such that for all . Thus for all , where . From scaling properties of stable densities, we get which is (26).
Now assume that . Using (29) and the fact that is radially decreasing, we may write

3. Main Results

Now, we are ready to announce and prove our main results.

Local existence of a classical nonnegative solution of (1)–(3) on a maximal interval of existence is obtained as usual (see, e.g., [22]).

Theorem 5. Assume ,   a.e. on . Let the nonlinearity satisfy (12), (14), and the polynomial growth condition (13). Then problem (1)–(3) admits a nonnegative classical solution on .

Proof. First, as and satisfies condition (14), we have and .
In view of the maximum principle, we have the estimate
Case  1 . From (1) and (2), we have which can alternatively be rewritten as Now, we use the duality argument. By multiplying (33) throughout by , the solution of (15)-(16), and integrating by parts over , we obtain or Using Lemma 2, we have Making use of inequality (24) together with , , and , we obtain Using estimates (36) and (37), we have Now, we have the estimates Finally, we have thanks to the above inequalities.
Since is arbitrarily nonnegative in and , therefore it follows by duality that Therefore, for all , the -norm of and remains finite on . From the polynomial growth assumption on the nonlinearity, it follows that is also in for all . If we take , we deduce that : This implies that .
Case  2 . This case can be treated in the same way by making use of inequality (20) with .

The next theorem deals with the “no growth” restriction on .

Theorem 6. Assume and , a.e. on . Let the nonlinearity satisfy (12) and (14). Then problem (1)–(3) has a classical solution on .

Proof. Let be the semigroup generated by on . Then we have where , or .
From (26), we have Using Lemma 1 for , we obtain So is bounded for any finite , whereupon the solution is global.

Remark 7. Our results remain valid when the reaction terms in the first equation and in the second equation satisfy where and are nonnegative constants.

Appendix

Here we present the result of Martin and Pierre [18] concerning the determination of the bound on the component of the system where satisfies hypotheses (12) and (14).

Theorem 8. Assume Then (A.1) has a classical solution on .

Proof. For , we can write via the semigroups and , where It is not difficult to see that From (A.3)–(A.6) and , we deduce which provides a uniform -bound for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 21/34/GR. The authors, therefore, acknowledge with thanks the DSR technical and financial support.