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Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 4070–4077 Research Article

Iterative solution for nonlinear impulsive advectionreaction-diffusion equations Xinan Haoa,∗, Lishan Liua,b , Yonghong Wub a

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China.

b

Department of Mathematics and Statistics, Curtin University, Perth, WA6845, Australia. Communicated by Y. Yao

Abstract Through solving equations step by step and by using the generalized Banach fixed point theorem, under simple conditions, the authors present the existence and uniqueness theorem of the iterative solution for nonlinear advection-reaction-diffusion equations with impulsive effects. An explicit iterative scheme for the c solution is also derived. The results obtained generalize and improve some known results. 2016 All rights reserved. Keywords: Iterative solution, nonlinear advection-reaction-diffusion equations, impulse. 2010 MSC: 35K57, 35R12.

1. Introduction In this paper, we shall investigate the following nonlinear impulsive advection-reaction-diffusion equations ut (t, x) = F (t, x, u(t, x), ∇u(t, x), ∆u(t, x)), ∆u(t, x)|t=tk = Ik (u(tk , x)), u(0, x) = u0 (x),

0 < t < T < ∞, x ∈ Ω ⊂ Rm , t 6= tk ,

(1.1)

x ∈ Ω, k = 1, 2, · · · , p,

(1.2)

x ∈ Ω,

(1.3)

where u(t, x) ∈ RN , F (t, x, u, ∇u, ∆u) ∈ RN , and t ∈ J = [0, T ] is the time variable, the subscript ut denotes partial differentiation with respect to t, u0 (x) ∈ RN , u(t, x) ∈ C 1 (0, T ) × C 2 (Ω) and continuous in ∗

Corresponding author Email addresses: [email protected] (Xinan Hao), [email protected] (Lishan Liu), [email protected] (Yonghong Wu) Received 2016-03-07

X. Hao, L. Liu, Y. Wu, J. Nonlinear Sci. Appl. 9 (2016), 4070–4077

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[0, T ], u0 (x) ∈ C 2 (Ω), ∆ is the Laplacian operator, ∇ is the gradient operator and Ω is a bounded spatial region. 0 < t1 < t2 < · · · < tk < · · · < tp < T, F ∈ C(J × Rm × RN × RN × RN , RN ), Ik ∈ C(RN , RN ) (k = − 1, 2, · · · , p). ∆u(t, x)|t=tk denotes the jump of u(t, x) at t = tk , i.e., ∆u(t, x)|t=tk = u(t+ k , x) − u(tk , x), where + − u(tk , x) and u(tk , x) represent the right and left limits of u(t, x) at t = tk , respectively. Advection-reaction-diffusion equations are used to simulate a variety of different phenomena, from mathematical biology to physics and engineering. Diffusion, advection and reaction respectively refer to those terms in the partial differential equations involving second, first and zero order derivatives of the unknowns with respect to the spatial variables. Equation (1.1) governs a large number of phenomena arising in chemical engineering, population dynamics, biology, physiology, combustion, ecology, chemotaxis, etc. [2, 5, 9, 21]. For example, in combustion and heat and mass transfer, u(t, x) may represent either the species concentrations or the temperature [21]. Impulsive differential equations arise naturally from a wide variety of applications, such as spacecraft control, inspection processes in operations research, drug administration and threshold theory in biology. Over the past decade, a significant advance in the theory of impulsive systems has been achieved. For the basic theory and recent development, the reader is referred to [7, 11, 12, 18] and the references therein. Over the last couple of decades, the existence, uniqueness, qualitative properties, and stability properties of solutions have been extensively studied for nonlinear advection-reaction-diffusion equations, see [3, 6, 8, 13, 15, 17, 19, 20, 22, 23]. In the special cases where F does not possess advection term, under several possible assumptions on the nonlinearity, the existence, uniqueness, stability properties of the special solutions and influence on the dynamics of the problems have been investigated in [19, 20, 22]. Recently, Ramos [16] presented an iterative method for solving nonlinear advection-reaction-diffusion equations without impulses and proved its convergence. The method was formulated in terms of a Picard operator and made use of Banach fixed-point theorem. In spite of the abundant literature on initial value problem for nonlinear advection-reaction-diffusion equations, there are few references dealing with this kinds of problems with impulses. The aim of this paper is to develop some theories of nonlinear advection-reaction-diffusion equations with impulsive terms. Motivated by the works [14], through solving equations step by step and by using the generalized Banach fixed point theorem (see [4]), we prove the existence and uniqueness solution for impulsive problem (1.1)-(1.3), and derive an approximation sequence of the solution which is explicitly expressed. Our results improve and generalize related results in [16] to some degree. The rest of the paper is organized as follows: In Section 2, we give some preliminaries to be used in the next section. The main results is formulated and proved in Section 3. 2. Preliminaries Let J 0 = J \ {t1 , t2 , · · · , tp }, J0 = [0, t1 ], J1 = (t1 , t2 ], · · · , Jp−1 = (tp−1 , tp ], Jp = (tp , T ]. We introduce a Banach space as follows [1, 10]. Let P C(J) = {u : for x ∈ Ω, u(·, x) is a map from J into L2 (Ω) such that u(t, x) is continuous at t 6= tk , left continuous at t = tk and its right limit at t = tk exists for k = 1, 2, · · · , p}, and the vector function spaces  SP C(J, Ω) = u = (u1 , u2 , · · · , uN )|ui ∈ P C(J) × L2 (Ω), i = 1, 2, · · · , N ,  H1 (J, Ω) = u = (u1 , u2 , · · · , uN )|ui ∈ C 1 (J) × L2 (Ω), i = 1, 2, · · · , N ,  H2 (J, Ω) = u = (u1 , u2 , · · · , uN )|ui ∈ C(J) × L2 (Ω), i = 1, 2, · · · , N . As in [10, 11], it is easily shown that SP C is a Banach space with the norm kukSP C = supt∈J ku(t, x)k, hP R i1 N 2 (t, x)dx 2 . where ku(t, x)k = u i=1 Ω i We need the following lemma in this paper. Lemma 2.1 ([14]). Suppose 0 < θ < 1, h > 0 are constants, let S = θn + Cn1 θn−1 h +

Cn2 θ2 h2 hn + ··· + , 2! n!

n ∈ N,

X. Hao, L. Liu, Y. Wu, J. Nonlinear Sci. Appl. 9 (2016), 4070–4077 then

 S≤o

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1

(n → +∞)

ns+1

for any real constant

s > 0.

3. Main results In this section, we give the main results of our paper. Theorem 3.1. If the following condition (H) is satisfied, (H) There exists a Lebesgue integrable nonnegative function q ∈ L2 (J, R+ ) such that kF (t, x, u, ∇u, ∆u) − F (t, x, v, ∇v, ∆v)k ≤ q(t)ku − vk, for any u, v ∈ SP C(J, Ω), (t, x) ∈ J × Ω. Then problem (1.1)-(1.3) has a unique solution η(t, x) ∈ SP C(J, Ω) ∩ H1 (J 0 , Ω) which can be written by  η0 (t, x),     η (t, x), 1 η(t, x) =  · ·· ,    ηp (t, x),

t ∈ J0 , t ∈ J1 , ··· , t ∈ Jp , x ∈ Ω.

Moreover, for any y0 ∈ SP C(J, Ω), the iterative sequence {yn }    (A0 y(n−1)0 )(t, x),    (A y 1 (n−1)1 )(t, x), yn (t, x) = (Ayn−1 )(t, x) =  ··· ,     (Ap y(n−1)p )(t, x),

defined by t ∈ J0 , t ∈ J1 , ··· , t ∈ Jp , x ∈ Ω, n = 1, 2, · · ·

converges uniformly to η(t, x) on (t, x) ∈ J × Ω, where  y (t, x), t ∈ J0 ,    (n−1)0  y t ∈ J1 , (n−1)1 (t, x), yn−1 (t, x) =  ··· , ··· ,     v(n−1)p (t, x), t ∈ Jp , x ∈ Ω,

 A ,   0  A , 1 A=  · · · ,    Ap ,

and A0 , Ai (i = 1, · · · , p) are defined by Z (A0 y(n−1)0 )(t, x) = u0 (x) +

t


F s, x, y(n−1)0 (s, x), ∇y(n−1)0 (s, x), ∆y(n−1)0 (s, x) ds,

t ∈ J0 ,

0

(Ai y(n−1)i )(t, x) =Ii (ηi−1 (ti , x)) + ηi−1 (ti , x) Z t
+ F s, x, y(n−1)i (s, x), ∇y(n−1)i (s, x), ∆y(n−1)i (s, x) ds,

t ∈ Ji .

ti

Remark 3.2. Comparing conditions (H1 ) − (H3 ) of paper [16] with (H) of this paper, we do not require (H1 ) and (H3 ), our condition (H) is weaker and more general than (H2 ), and by means of a completely different method with [16], we have proven the existence of a unique solution for impulsive problem (1.1)-(1.3). Our result in essence improves and generalizes related results in [16] to some degree. Remark 3.3. It is value to point out that the iterative sequences {yn } are expressed explicitly, which is an important improvement compared with those in the above mentioned papers.

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Proof. Our proof is divided into four steps. For 0 < ε < T1 , from the property of Lebesgue integrable functions, there exists a continuous function RT ψ(t) in [0, T ] such that 0 |q 2 (t) − ψ(t)|dt < ε. Let M = supt∈J |ψ(t)|, evidently, 0 ≤ M < +∞. Step 1. We consider the following initial value problem to nonlinear advection-reaction-diffusion equation: ( ut (t, x) = F (t, x, u(t, x), ∇u(t, x), ∆u(t, x)), t ∈ J0 , x ∈ Ω, (3.1) u(0, x) = u0 (x). It is well known that u ∈ SP C(J0 , Ω) ∩ H1 (J0 , Ω) is a solution of (3.1) if only if u ∈ H2 (J0 , Ω) is a solution of the following integral equation: Z t F (s, x, u(s, x), ∇u(s, x), ∆u(s, x))ds, t ∈ J0 , x ∈ Ω. u(t, x) = u0 (x) + 0

Define operator A0 by t

Z

F (s, x, u(s, x), ∇u(s, x), ∆u(s, x))ds,

(A0 u)(t, x) = u0 (x) +

t ∈ J0 , x ∈ Ω.

(3.2)

0

Clearly, A0 : H2 (J0 , Ω) → H2 (J0 , Ω). From (H) and (3.2) and Cauchy-Schwarz-Bunyakovski inequality, for u, v ∈ H2 (J0 , Ω), t ∈ J0 , x ∈ Ω, we have k(A0 u)(t, x) − (A0 v)(t, x)k2

2

Z t

= (F (s, x, u, ∇u, ∆u) − F (s, x, v, ∇v, ∆v))ds

=

0 N XZ

Z

(Fi (s, x, u, ∇u, ∆u) − Fi (s, x, v, ∇v, ∆v))ds

Ω

i=1

2

t

dx

0

 N Z  Z t X 2 ≤ t (Fi (s, x, u, ∇u, ∆u) − Fi (s, x, v, ∇v, ∆v)) ds dx

=

i=1 Ω N Z t X

0

Z

 (Fi (s, x, u, ∇u, ∆u) − Fi (s, x, v, ∇v, ∆v))2 dx ds

t

Ω i=1 0  Z Z N t X

≤T

0

i=1

=T

(Fi (s, x, u, ∇u, ∆u) − Fi (s, x, v, ∇v, ∆v)) dx ds 2

0

Z

t

q 2 (s)ku(s, x) − v(s, x)k2 ds Z t  Z t 2 ≤T |q (s) − ψ(s)|ds + |ψ(s)|ds ku − vk2SP C

≤T

0

0

≤T (ε + M t)ku − From (3.2) and (3.3), we obtain



(Fi (s, x, u, ∇u, ∆u) − Fi (s, x, v, ∇v, ∆v)) dx ds

Ω

kF (s, x, u, ∇u, ∆u) − F (s, x, v, ∇v, ∆v))k2 ds

=T



Ω

Z tX N Z 0 i=1 Z t

2

0

vk2SP C .

(3.3)

X. Hao, L. Liu, Y. Wu, J. Nonlinear Sci. Appl. 9 (2016), 4070–4077 k(A20 u)(t, x) − (A20 v)(t, x)k2

Z t

2

= (F (s, x, A0 u, ∇(A0 u), ∆(A0 u)) − F (s, x, A0 v, ∇(A0 v), ∆(A0 v))ds

0 Z t ≤T kF (s, x, A0 u, ∇(A0 u), ∆(A0 u)) − F (s, x, A0 v, ∇(A0 v), ∆(A0 v)k2 ds 0 Z t q 2 (s)k(A0 u)(s, x) − (A0 v)(s, x)k2 ds ≤T 0 Z t q 2 (s)(ε + M s)dsku − vk2SP C ≤T 2 0 Z t  Z t |q 2 (s) − ψ(s)|(ε + M s)ds + |ψ(s)|(ε + M s)ds ku − vk2SP C ≤T 2 0 0   2 t2 M ≤T 2 ε(ε + M t) + M εt + ku − vk2SP C . 2

4074

(3.4)

In the following, by the method of mathematical induction, for any positive integer n and (t, x) ∈ J0 × Ω, we will prove that k(An0 u)(t, x) − (An0 v)(t, x)k2   (M t)j (M t)n ≤T n εn + Cn1 εn−1 (M t) + · · · + Cnj εn−j + ··· + ku − vk2SP C , j! n!

(3.5)

n! where Cnj = j!(n−j)! , j! = j · (j − 1) · · · 3 · 2 · 1. When n = 1, (3.5) holds by (3.3). For n = 2, (3.5) holds by (3.4). Suppose (3.5) holds for n = k, that is, for any (t, x) ∈ J0 × Ω,   j (M t)k j k−j (M t) k k 2 k k 1 k−1 k(A0 u)(t, x) − (A0 v)(t, x)k ≤ T ε + Ck ε (M t) + · · · + Ck ε + ··· + ku − vk2SP C . j! k! j = Ckj + Ckj−1 , for any (t, x) ∈ J0 × Ω, one has Then, by (H), (3.2), (3.3), and applying formula Ck+1 k+1 2 k(Ak+1 0 u)(t, x) − (A0 v)(t, x)k

Z t

2

k k k k k k

= (F (s, x, A u, ∇(A u), ∆(A u)) − F (s, x, A v, ∇(A v), ∆(A v))ds 0 0 0 0 0 0

0 Z t ≤T kF (s, x, Ak0 u, ∇(Ak0 u), ∆(Ak0 u)) − F (s, x, Ak0 v, ∇(Ak0 v), ∆(Ak0 v))k2 ds 0 Z t ≤T q 2 (s)k(Ak0 u)(s, x) − (Ak0 v)(s, x)k2 ds 0   Z t j (M s)k j k−j (M s) 2 k 1 k−1 k+1 ≤T q (s) ε + Ck ε (M s) + · · · + Ck ε + ··· + dsku − vk2SP C j! k! 0 Z t   j (M s)k j k−j (M s) k+1 2 k 1 k−1 ≤T |q (s) − ψ(s)| ε + Ck ε (M s) + · · · + Ck ε + ··· + ds j! k! 0    Z t j (M s)k j k−j (M s) k 1 k−1 + |ψ(s)| ε + Ck ε (M s) + · · · + Ck ε + ··· + ds ku − vk2SP C j! k! 0    j (M t)k j k−j (M t) k+1 k 1 k−1 ≤T ε ε + Ck ε (M t) + · · · + Ck ε + ··· + j! k!   j+1 2 (M t)k+1 j k−j (M t) k 1 k−1 (M t) + ε M t + Ck ε + · · · + Ck ε + ··· + ku − vk2SP C 2! (j + 1)! (k + 1)!   j (M t)k+1 j k+1 k+1 1 k k−j+1 (M t) =T ε + Ck+1 ε (M t) + · · · + Ck+1 ε + ··· + ku − vk2SP C . j! (k + 1)!

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Hence, (3.5) holds for n = k + 1. Therefore, for any positive integer n, denote θ = T ε, h = M T 2 , we have   j hn j k−j h n n 2 n 1 n−1 (3.6) kA0 u − A0 vkSP C ≤ θ + Cn θ h + · · · + Ck θ ku − vk2SP C . + ··· + j! n! Consequently, Lemma 2.1 and (3.6) imply that for any real constant s > 0, there exists a positive integer n0 such that for any u, v ∈ H2 (J0 , Ω), kAn0 u − An0 vk2SP C ≤ and kAn0 u − An0 vkSP C ≤

1 ku − vk2SP C , ns+1

∀n > n0 ,

1

ku − vkSP C , ∀n > n0 . n So An0 is a contraction operator on H2 (J0 , Ω). By the generalized Banach Contraction Theorem, we conclude that A0 has only one fixed point η0 ∈ H2 (J0 , Ω). This implies that (3.1) has a unique solution η0 ∈ SP C(J0 , Ω) ∩ H1 (J0 , Ω) such that ( (η0 )t (t, x) = F (t, x, η0 (t, x), ∇η0 (t, x), ∆η0 (t, x)), t ∈ J0 , x ∈ Ω, (3.7) η0 (0, x) = u0 (x). s+1 2

Step 2. We consider the following nonlinear advection-reaction-diffusion equation: ( ut (t, x) = F (t, x, u(t, x), ∇u(t, x), ∆u(t, x)), t ∈ J1 , x ∈ Ω, u(t+ 1 , x) = I1 (η0 (t1 , x)) + η0 (t1 , x).

(3.8)

It is easy to prove that u ∈ SP C(J1 , Ω) ∩ H1 (J10 , Ω) is a solution of (3.8) if only if u ∈ H2 (J10 , Ω) is a solution of the following integral equation: Z t u(t, x) = I1 (η0 (t1 , x)) + η0 (t1 , x) + F (s, x, u(s, x), ∇u(s, x), ∆u(s, x))ds, t ∈ J1 , x ∈ Ω. t1

Let Z

t

F (s, x, u(s, x), ∇u(s, x), ∆u(s, x))ds,

(A1 u)(t, x) = I1 (η0 (t1 , x)) + η0 (t1 , x) +

t ∈ J1 , x ∈ Ω.

(3.9)

t1

Clearly, A1 : H2 (J1 , Ω) → H2 (J1 , Ω). From (H) and (3.9), for u, v ∈ H2 (J1 , Ω), we get k(A1 u)(t, x) − (A1 v)(t, x)k2

Z t

2

= (F (s, x, u(s, x), ∇u(s, x), ∆u(s, x)) − F (s, x, v(s, x), ∇v(s, x), ∆v(s, x)))ds

t1 t

Z

kF (s, x, u, ∇u, ∆u) − F (s, x, v, ∇v, ∆v))k2 ds

≤T t1 Z t

q 2 (s)ku(s, x) − v(s, x)k2 ds

≤T t

Z1 ≤T

t

|q 2 (s) − ψ(s)|ds +

0

Z

t

 |ψ(s)|ds ku − vk2SP C

0

≤T (ε + M t)ku − vk2SP C . Similar to the proof of Step 1, An1 is a contraction operator on H2 (J1 , Ω). By the Banach Contraction Theorem, we conclude that A1 has only one fixed point η1 ∈ H2 (J1 , Ω), that is (3.8) has a unique solution η1 ∈ SP C(J1 , Ω) ∩ H1 (J10 , Ω) such that ( (η1 )t (t, x) = F (t, x, η1 (t, x), ∇η1 (t, x), ∆η1 (t, x)), t ∈ J1 , x ∈ Ω, (3.10) η1 (t+ 1 , x) = I1 (η0 (t1 , x)) + η0 (t1 , x).

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Step 3. For i = 2, 3, · · · , p, we repeat the above procedure, then the following problem: ( ut (t, x) = F (t, x, u(t, x), ∇u(t, x), ∆u(t, x)), t ∈ Ji , x ∈ Ω, u(t+ i , x) = Ii (ηi−1 (ti , x)) + ηi−1 (ti , x), has a unique solution ηi (t, x) ∈ SP C(Ji , Ω) ∩ H1 (Ji0 ) such that ( (ηi )t (t, x) = F (t, x, ηi (t, x), ∇ηi (t, x), ∆ηi (t, x)), ηi (t+ i , x)

= Ii (ηi−1 (ti , x)) + ηi−1 (ti , x).

t ∈ Ji , x ∈ Ω,

(3.11)

Let  η0 (t, x),     η (t, x), 1 η(t, x) =  ··· ,    ηp (t, x),

t ∈ J0 , t ∈ J1 , ··· ,

(3.12)

t ∈ Jp , x ∈ Ω.

Then, from (3.7), (3.10)-(3.12), η(t, x) ∈ SP C(J, Ω) ∩ H1 (J 0 , Ω) is unique solution of impulsive problem (1.1)-(1.3). Step 4. For any y0 ∈ SP C(J, Ω), let yn = Ayn−1 , where  (A0 y(n−1)0 )(t, x), t ∈ J0 ,      (A y t ∈ J1 , 1 (n−1)1 )(t, x), (Ayn−1 )(t, x) =  ··· , ··· ,     (Ap y(n−1)p )(t, x), t ∈ Jp , x ∈ Ω,    y(n−1)0 (t, x), t ∈ J0 ,   y t ∈ J1 , (n−1)1 (t, x), yn−1 (t, x) =  ··· , ··· ,     y(n−1)p (t, x), t ∈ Jp , x ∈ Ω. From the proof of Step 1-Step 3, it is easy to prove that the iterative convergence theorem holds. Thus, we complete the proof of Theorem 3.1. Acknowledgment The authors were supported financially by the National Natural Science Foundation of China (11501318, 11371221) and the Natural Science Foundation of Shandong Province of China (ZR2015AM022). References [1] R. P. Agarwal, M. Meehan, D. O’Regan, Fixed point theory and applications, Cambridge University Press, Cambridge, (2001). 2 [2] D. G. Aronson, H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), 5–49. Lecture Notes in Math., Vol. 446, Springer, Berlin, (1975). 1 [3] Z. Bu, Z. Wang, N. Liu, Asymptotic behavior of pulsating fronts and entire solutions of reaction-advection-diffusion equations in periodic media, Nonlinear Anal. Real World Appl., 28 (2016), 48–71. 1 [4] K. Deimling, Nonlinear functional analysis, Springer, Berlin, (1985). 1 [5] P. C. Fife, Mathematical aspects of reacting and diffusing systems, Springer, Berlin, (1979). 1 [6] J. Ge, K. I. Kim, Z. Lin, H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486–5509. 1 [7] X. Hao, L. Liu, Y. Wu, Positive solutions for second order impulsive differential equations with integral boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 101–111. 1

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