Global existence of solutions to a nonlinear anomalous diffusion system

Accepted Manuscript Global existence of solutions to a nonlinear anomalous diffusion system Bashir Ahmad, Ahmed Alsaedi, Mokhtar Kirane PII: DOI: Reference:

S0893-9659(16)30085-4 http://dx.doi.org/10.1016/j.aml.2016.03.006 AML 4970

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Applied Mathematics Letters

Received date: 15 February 2016 Revised date: 7 March 2016 Accepted date: 8 March 2016 Please cite this article as: B. Ahmad, A. Alsaedi, M. Kirane, Global existence of solutions to a nonlinear anomalous diffusion system, Appl. Math. Lett. (2016), http://dx.doi.org/10.1016/j.aml.2016.03.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Global existence of solutions to a nonlinear anomalous diffusion system Bashir Ahmad1,∗ , Ahmed Alsaedi1 , Mokhtar Kirane2

1

NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2

D´e de Math´ematiques, Universit´e de La Rochelle, Avenue M. Cr´epeau, 17042 La Rochelle Cedex, France

E-mail: bashirahmad [email protected] (BA), [email protected] (AA), [email protected] (MK) Abstract A nonlinear system with different anomalous diffusion terms is considered. The existence of global positive solutions is proved.

Key words: Fractional Laplacian; global existence. Mathematical Subject Classification: 35A01, 35R11.

1

Introduction

In this paper, we consider the following system of nonlinear fractional in space reactiondiffusion equations  ut (x, t) + ta (−∆)α u(x, t) = A(t) v r (x, t) us (x, t), x ∈ RN , t > 0, (1.1) vt (x, t) + tb (−∆)β v(x, t) = B(t) v p (x, t) uq (x, t), x ∈ RN , t > 0, for u > 0, v > 0, equipped with the initial conditions u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ RN , where the initial data

∗

 u0 , v0 ∈ CLB (RN ) := φ ∈ C(RN ) : φ ≥ c > 0 ,

Corresponding author.

(1.2)

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B. Ahmad, A. Alsaedi, M. Kirane

are given functions, a > 0, b > 0, s < 0, p < 0, r < 1 − p and q < 1 − s. The functions A and B are such that A(t) ≥ c0 tk and B(t) ≥ c1 tl where c0 > 0, c1 > 0, k > 0 and l > 0. Here the nonlocal operator (−∆)µ , 0 < µ ≤ 1 (µ = α, β) stands for anomalous diffusion [11] and is defined, for any function u in the Schwartz space, by the Fourier transform pair F and F −1
(−∆)µ u(x) = F −1 |ξ|2µ F u(ξ) (x); it has the Reisz representation µ

(−∆) u(x) = CN,µ P V

Z

RN

u(x) − u(y) dy, |x − y|N +2µ

where CN,µ is a normalizing constant depending only on N and µ [12], such that limµ→1− (−∆)µ φ = −∆φ. We will show that system (1.1) − (1.2) admits globally bounded solutions relying on a comparison argument. Heat equations and systems have received a great attention concerning global existence and blow-up of solutions; they are well documented and we only refer to the important books [18] , [17], [19], and [14]. However, the work on parabolic fractional differential equations is scarce and it started with the paper of Nagasawa and Sirao [13] who used a probabilistic treatment of blowing-up solutions to equations with fractional powers of the Laplacian of the form ut + (−∆)α u = c(x) |u|p , for a certain positive function c(x). Sugitani [20] treated the same equation with c(x) = 1, while Kobayashi [10] discussed a more general equation. Concerning systems of fractional in space differential equations, we can mention the papers [2], [7] and [15]. Kirane et al. [9] investigated a further extension to fractional in time and space systems. The system (1.1) when it pops up with positive exponents for the nonlinear terms appears in combustion theory [18] while the cases for s = p = 0, r = −2 and q = −2 appear in the mathematical analysis of micro-electromechanical systems (MEMS) and have important applications such as accelerometers for airbag deployment in cars, inject printer heads, etc., for instance, see [5], [16] and [21]. Our system has both positive and negative exponents even though the negative exponents do not pop up as in the MEMS system.

2

Preliminaries

First of all, we recall some fundamental facts. Let Sα (t) be the semi-group associated with the heat equation ut + (−∆)α u = 0, 0 < α ≤ 1, t > 0, x ∈ RN .

Global existence of solutions

3

It is known that Sα (t) defined by Sα (t)(x) =

1 (2π)

N 2

Z

α

eixξ−t|ξ| dξ RN

satisfies the following properties: • Sα (t) ∈ L∞ (RN Z ) ∩ L1 (RN );

• Sα (t) ≥ 0 and

RN

Sα (t)dx = 1, x ∈ RN , t > 0,

and the following estimates: • kSα (t) ∗ u0 kp ≤ ku0kp , u0 ∈ Lp (RN ), 1 ≤ p ≤ ∞, t > 0; N 1 1 • kSα (t) ∗ u0 kq ≤ ct− α ( p − q ) ku0 kp ; 1 1 N • k∇Sα (t)kq ≤ ct− α (1− q )− α for any u0 ∈ Lp (RN ), 1 ≤ p < q ≤ ∞, t > 0. Let {U(t, s)t>s≥0 } and {V(t, s)t>s≥0 } be the evolution families on CB (RN ) that describe the solutions Cauchy problem for the families of generators  bto theβhomogeneous a α {t (−∆) }t≥0 and t (−∆) t≥0 , respectively. We know from [6] that   a+1 t − σ a+1 , t≥σ≥0 U(t, σ) = Sα a+1

and V(t, σ) = Sβ



tb+1 − σ b+1 b+1



,

t ≥ σ ≥ 0,

where {Sα (t)}t≥0 and {Sβ (t)}t≥0 are the semigroups with infinitesimal generators ∆α and ∆β , respectively. We also need the following lemma due to A. C´ordoba, D. C´ordoba [3] in the forthcoming analysis. Lemma 2.1 Let Φ be a C 2 convex and increasing function such that Φ(0) = 0. Let w ∈ C02 (RN ) and let 0 ≤ s ≤ 2. Then Φ′ (w)(−∆)s w ≥ (−∆)s Φ(w)

(2.1)

holds pointwise a.e in RN .

3

Main result

Letting f (t, u, v) = A(t)v r us and g(t, u, v) = B(t)v p uq , we rewrite problem (1.1) as  ut (x, t) + ta (−∆)α u(x, t) = f (t, u(x, t), v(x, t)), (3.1) vt (x, t) + tb (−∆)β v(x, t) = g(t, u(x, t), v(x, t)),

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B. Ahmad, A. Alsaedi, M. Kirane

in RN × (0, +∞), supplemented with the initial data (1.2). It is natural to associate the system (3.1) with the corresponding pair of integral equations Z t u(t, x) = U(t, 0)u0 (x) + U(t, s)f (s, u(x, s), v(x, s)) ds, t > 0, x ∈ RN , (3.2) 0

v(t, x) = V(t, 0)v0 (x) +

Z

0

t

V(t, s)g(s, u(x, s), v(x, s)) ds,

t > 0, x ∈ RN .

(3.3)

We mention that for u0 ∈ CLB (RN ) ∩ H α (RN ), v0 ∈ CLB (RN ) ∩ H β (RN ), the system (1.1) − (1.2) has a nonnegative local solution (u, v) ∈ C([0, Tmax ); H α (RN )) × C([0, Tmax ); H β (RN )) which can be obtained by using the Banach fixed point theorem (see [6]) as the reaction terms are locally Lipschitzian. Let us now state and prove our main theorem. Theorem 3.1 Let u0 ∈ CLB (RN ) ∩ H α (RN ), v0 ∈ CLB (RN ) ∩ H β (RN ). Then the solution of (1.1) − (1.2) is global and bounded. Proof. As u0 ≥ c > 0 and v0 ≥ c > 0, we obtain u ≥ c > 0 and v ≥ c > 0 by standard technique of parabolic systems. Next, we show that the solution (u, v) is bounded. Multiplying the first and second equations of (1.1) by θ uθ−1 and λ v λ−1 respectively, we obtain ( θ uθ−1ut (x, t) + ta θ uθ−1 (−∆)α u(x, t) = A(t) θ v r (x, t) us+θ−1(x, t), (3.4) λ v λ−1 vt (x, t) + tb λ v λ−1 (−∆)β v(x, t) = B(t) λ v p+λ−1(x, t) uq (x, t), for x ∈ RN , t > 0. If we set U = uθ and V = v λ , then ( s+θ−1 r v r us+θ−1 = V λ U θ , q p+λ−1 v p+λ−1 uq = V λ U θ . Now we choose θ and λ so that s + θ − 1 = 0 and p + λ − 1 = 0, which gives θ = 1 − s and λ = 1 − p. In consequence, we have ( r s+θ−1 r VλU θ = V 1−p , q q p+λ−1 V λ U θ = U 1−s . Now if θ ≥ 1 and λ ≥ 1, that is, s ≤ 0 and p ≤ 0, using the inequality of C´ordoba and C´ordoba [3], we get  θ uθ−1 (−∆)α u ≥ (−∆)α uθ , u ≥ 0, (3.5) λ v λ−1 (−∆)β v ≥ (−∆)β v λ , v ≥ 0.

Global existence of solutions Let us set γ =

5

r q < 1 and σ = < 1. Then the system (1.1) via (3.4) and 1−p 1−s

(3.5) leads to ( Ut (x, t) + ta (−∆)α U(x, t) ≤ θ A(t)V γ (x, t), x ∈ RN , t > 0,

Vt (x, t) + tb (−∆)β V (x, t) ≤ λ B(t)U σ (x, t), x ∈ RN , t > 0,

(3.6)

supplemented with the positive and bounded initial conditions U(x, 0) = uθ0 (x), V (x, 0) = v0λ (x), x ∈ RN . Thus we can write  Z t    U(x, t) ≤ U(t, 0)U(x, 0) + θ U(t, s)A(s)V γ (x, s) ds, Z0 t    V (x, t) ≤ V(t, 0)V (x, 0) + λ V(t, s)B(s)U σ (x, s) ds.

(3.7)

(3.8)

0

So the solution (U, V ) of problem (3.6) − (3.7) can be bounded componentwise by the solution (U , V ) of the system ( γ U t (x, t) + ta (−∆)α U(x, t) = θ A(t)V (x, t), x ∈ RN , t > 0, (3.9) σ V t (x, t) + tb (−∆)β V (x, t) = λ B(t)U (x, t), x ∈ RN , t > 0,

supplemented with the initial conditions U (x, 0) = U(x, 0) and V (x, 0) = V (x, 0), x ∈ RN . As γ < 1, σ < 1 and the initial data U (x, 0) and V (x, 0) are positive and bounded, the solutions to (3.9) are bounded as the nonlinearities are sub-linear. So the L∞ -norms of U and V remain bounded for any t > 0. But U = uθ and V = v λ , whereupon the solution of (1.1) − (1.2) is global.

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