On a Stochastic Coupled System of Reaction-Diffusion of Nonlocal Type
In this article we investigate the existence and uniqueness of weak solutions for a stochastic nonlinear parabolic coupled system of reaction-diffusion of nonlocal type, and with multiplicative white noise. An important result on the asymptotic behavior o
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1 Introduction In this paper, we study the following initial-boundary value problem involving a stochastic nonlinear parabolic system of nonlocal type Z 8 @W1 ˆ ˆ ut a u dx u D g1 .v/ C f1 .u; v/ ˆ ˆ @t ˆ D ˆ ˆ ˆ Z ˆ < @W2 vt a v dx v D g2 .u/ C f2 .u; v/ @t D ˆ ˆ ˆ ˆ ˆ ˆ .u.x; 0/; v.x; 0// D .u0 .x/; v0 .x// in D; ˆ ˆ ˆ : .u; v/ D .0; 0/ on @D 0; 1Œ ;
on
D 0; 1Π;
on
D 0; 1Π;
(1)
E.A. Coayla-Teran () IM – Universidade Federal da Bahia, Salvador, Brasil e-mail: [email protected] J. Ferreira UAG – Universidade Federal Rural de Pernambuco, Garanhuns, Brasil CMAF – Universidade de Lisboa, Lisboa, Portugal e-mail: [email protected] P.M.D. de Magalhães DM – Universidade Federal de Ouro Preto, Ouro Preto, Brasil e-mail: [email protected] H.B. de Oliveira FCT – Universidade do Algarve, Faro, Portugal CMAF – Universidade de Lisboa, Lisboa, Portugal e-mail: [email protected] C. Bernardin and P. Gonçalves (eds.), From Particle Systems to Partial Differential Equations, Springer Proceedings in Mathematics & Statistics 75, DOI 10.1007/978-3-642-54271-8__15, © Springer-Verlag Berlin Heidelberg 2014
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where D is an open bounded subset of Rn , n 1, with a smooth boundary @D. On the nonlocal function a D a.s/, we assume that: a is a Lipschitz-continuous function with Lipschitz constant denoted by LI
(2)
0 < p a.s/ P ;
(3)
where p and P are constants.
.W1 .t /; W2 .t //t2Œ0;1Œ is a two-dimensional real Wiener process, the maps fi W L2 .D/ L2 .D/ ! L2 .D/; gi W L2 .D/ ! L2 .D/; with i D 1; 2 satisfy the following conditions: kfi .z1 ; x1 / fi .z2 ; x2 /k2 J kz1 z2 k2 C kx1 x2 k2 8 z1 ; z2 ; x1 ; x2 2 L2 .D/I (4) fi .0; 0/ D 0I
(5)
kgi .z1 / gi .z2 /k2 K kz1 z2 k2 8 z1 ; z2 2 L2 .D/I
(6)
gi .0/ D 0 ;
(7)
where J and K are positive constants:
Moreover, the multiplicative white noise, represented by the random forcing term in the model, describes a state dependent random noise. g.t; u/ @W @t Remark 1. For simplicity we are considering reaction terms g1 and g2 depending only on one quantity: v or u, respectively. But we may as well consider their dependence on both quantities u and v, as long as they satisfy to the corresponding version of the assumption (4). The consideration of cross reactions g1 .v/ and g2 .u/ is only to emphasize the existence of coupling also on the reaction terms. For the last several decades, various types of equations have been employed as some mathematical model describing physical, chemical, biological and ecological systems. Among them, the most successful and crucial one is the following model of semilinear parabolic partial differential equation @u Au f .u/ D 0; @t
(8)
where f W Rn ! Rn is a nonlinear function, and A is an n n real matrix. In [15] it was considered the reaction-diffusion equation (8), where A is an n n real matrix and f W Rn ! Rn is a C 2 function. There, it was studied the exponential decay for some cases. In the literature, most authors assume that the diffusion matrix
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