GLOBAL DYNAMICS OF p -LAPLACIAN REACTION-DIFFUSION EQUATIONS WITH APPLICATION TO VIROLOGY

. In this paper, we propose a method to investigate the global stability of reaction-diffusion equations involving the p-Laplacian operator with and without delay. The proposed method is based on the direct Lyapunov method which consists to construct an appropriate Lyapunov functional. Furthermore, the method is applied to two biological systems from virology one without delay and the other with both delays in the infection and viral production.


INTRODUCTION
The p-Laplacian operator is a generalization of the classical Laplacian operator. It has been used to study some turbulent fluids through porous media. For example, Diaz and De Thelin [1] focused on a nonlinear parabolic problem arising in some models related to turbulent flows.
Ahmed and Sunada [2] dealt with nonlinear flow in porous media. Volker [3] investigated nonlinear flow in porous media by finite elements. In addition, the p-Laplacian operator is also used in the modeling of non-Newtonian fluids [4,5,6].
Reaction-diffusion equations involving p-Laplacian operator have been studied by many authors. For instance, Gmira [7] proved the existence of nontrivial solution of the quasilinear parabolic equations under some conditions. Kamin and Vázquez [8] studied the existence and uniqueness of singular solutions for some nonlinear parabolic equations. Peletier and Wang [9] established the existence of a very singular solution of a quasilinear degenerate diffusion equation with absorption. Bettioui and Gmira [10] proved, under suitable conditions on the parameters, the existence, uniqueness as well as the qualitative behavior of radial solutions of a degenerate quasilinear elliptic equation in IR n . The results presented in [10] have been extended by Bidaut-Véron [11].
On the other hand, the stability of reaction-diffusion equations with the classical Laplacian operator has been investigated by several researchers. However, to our knowledge there is no work for the global stability of reaction-diffusion equations involving the p-Laplacian operator.
Therefore, the main purpose of this paper is to extend the method presented in [12] in order to study the global stability of p-Laplacian reaction-diffusion equations with and without delay.
To do this, Section 2 is devoted to the description of the extended method. Finally, Section 3 deals with an application in virology.

DESCRIPTION OF THE METHOD
Let u = (u 1 , . . . , u m ) be the non-negative solution of the ordinary differential equation where f : IR m −→ IR m is a C 1 function.
Let Ω be a bounded domain in IR n with smooth boundary ∂ Ω and D = diag(d 1 , . . . , d m ) with Suppose u * is a non-negative equilibrium of (1), then u * is also a spatially homogeneous steady state of the following reaction-diffusion system with Neumann boundary condition where p ≥ 2, ∆ p u = div |∇u| p−2 ∇u is the p-Laplacian operator and ∇u is the gradient of u.
Let V (u) be a C 1 function defined on some domain in IR m + . If we put On the other hand, we have Hence, Therefore, we construct the function V such that In the literature, many authors like in [13,14] constructed the explicit Lyapunov functions of the form : In this case, we have Thus, We summarize the above results in the following proposition.
(i) If the Lyapunov function V for the ordinary differential equation (1) verifies the condition (6), then the function W defined by (3) is a Lyapunov functional for the reactiondiffusion system (2).
(ii) If the Lyapunov function V for the ordinary differential equation (1) is of the form described by (7), then W is a Lyapunov functional for the reaction-diffusion system (2).
As in [12], consider the following delayed reaction-diffusion equation where τ ≥ 0, the function u t is defined on Ω × [−τ, 0] by u t (x, θ ) = u(x,t + θ ) and g is a functional of u, u t . In this case, the time derivative of the function W defined by (3) along the positive solution of (9) satisfies Therefore, Like in [12], the integrands of the first and second terms are already calculated. By means idea of Kajiwara et al. [15], the integrand of the third term can be modified to show the negativeness of the time derivative of a Lyapunov function for (9).

APPLICATION TO VIROLOGY
In this section, we apply the method described in the previous section to a virological system with and without delay.
Example 1: Consider the following reaction system: where the infected target cells (U) are produced at a constant rate λ , die at a rate dU and become infected by virus at a rate βVU. Infected cells (I) die at rate aI. Free virus (V ) is produced by infected cells at a rate kI and decays at a rate µV .
To model the mobility of virus, we propose the following system : where U(x,t), I(x,t) and V (x,t) denote the densities of infected target cells, infected cells and free virus at position x and time t, respectively. In addition, the parameter d V is the diffusion coefficient.
We consider the system (12) with Neumann boundary condition ∂V ∂ ν = 0 on ∂ Ω × (0, +∞), and initial conditions Clearly, the system (12) has an infection-free equilibrium Q 0 (U 0 , 0, 0) with U 0 = λ d and the basic reproduction number is given by In addition, system (12) has another equilibrium named chromic equilibrium of the form Let u = (U, I,V ) be a solution of (11). To establish the stability of the infection-free equilibrium Q 0 , we consider the following Lyapunov functional By a simple computation, we find By applying Proposition 2.1, we construct a Lyapunov functional for reaction-diffusion system (12), as follows Then As R 0 ≤ 1, we have dW 0 dt ≤ 0. So, W 0 is a Lyapunov functional of (12) at equilibrium Q 0 when For R 0 > 1, we consider the following Lyapunov functional Similar calculations give Hence, Then dW 1 dt ≤ 0. Therefore, W 1 is a Lyapunov functional of (12) at equilibrium Q * .
Example 2: To describe both delays in the infection and viral production as in [16], system (11) becomes Here, the first delay τ 1 is the time needed for infected cells to produce virions after viral entry.
We assume that virus production lags by a delay τ 1 behind the infection of a cell. This implies that recruitment of virus-production cells at time t is given by the number of cells that were newly infected at time t − τ 1 and are still alive at time t. We assume that the death rate for The system (14) has an infection-free equilibrium Q 0 ( λ d , 0, 0) and the basic reproduction number On the other hand, system (14) has another equilibrium Q * ( U * , I * , V * ) where For u = (U, I,V ) a solution of (13), consider the following Lyapunov functional and let A similar calculations as in [16] , we get If R 0 ≤ 1, then d W 0 dt ≤ 0 and the disease-free equilibrium Q 0 is globally asymptotically stable.
When R 0 > 1, we consider the following Lyapunov functional Hence, If we put W 1 = Ω H 1 (u(x,t))dx, we obtain Thus, d W 1 dt ≤ 0 and W 1 is a Lyapunov functional of (14) at Q * when R 0 > 1.