THE GENERALIZED BURGERS EQUATION WITH AND WITHOUT A TIME DELAY

We consider the generalized Burgers equation with and without a time delay when the boundary conditions are periodic with period 2π. For the generalized Burgers equation without a time delay, that is, ut = νuxx − uux + u+ h(x), 0 < x < 2π, t > 0, u(0, t) = u(2π, t), u(x,0)= u0(x), a Lyapunov function method is used to show boundedness and uniqueness of a steady state solution and global stability of the equation. As for the generalized time-delayed Burgers equation, that is, ut(x, t)= νuxx(x, t)−u(x, t− τ)ux(x, t) + u(x, t), 0 < x < 2π, t > 0, u(0, t)= u(2π, t), t > 0, u(x,s)= u0(x,s), 0 < x < 2π, −τ ≤ s≤ 0, we show that the equation is exponentially stable under small delays. Using a pseudospectral method, we present some numerical results illustrating and reinforcing the analytical results.


Introduction
Recently, the generalized Burgers equation has gotten a lot of attention and interest from both the engineering and the mathematical communities to model several problems including but not limited to the control of turbulent flow [2,9], the excitation of long water waves by a moving pressure distribution [1], the dispersal of a population [32], and the behavior of the flame front interface under physical assumption [29].Rakib and Sivashinsky [29] derived a nonlinear evolution equation as a model for the flame front interface: y x (0,t) = 0, y x (1,t) = 0, y(x,0) = y 0 (x), (1.3) where ν > 0 is a small positive constant.Later on, Sun and Ward [33] studied (1.2) by reformulating it in terms of the slope u(x,t) = −y x (x,t), which yields u t = νu xx − uu x + u, 0< x < 1, t > 0, (1.4) u(0,t) = u(1,t) = 0, u(x,0) = u 0 (x). (1.5) They showed that for ν 1 with a certain class of initial conditions, the solution will have a metastable behavior.Generally, a solution is called metastable if the change of its motion can be noticed only on very long-time intervals [13].
In this paper, we study the behavior of the solution of (1.1) on [0,2π], with periodic boundary conditions, and different values of h.It should be noted that the case where m = 1 and h = 0 in (1.1) reduces to (1.4).Also, we investigate the dynamical behavior of the generalized Burgers equation if a time delay τ is introduced in the convective term.That is, u t (x,t) = νu xx (x,t) − u(x,t − τ)u x (x,t) + u(x,t), 0 < x < 2π, t > 0, (1.6) u(0,t) = u(2π,t), t > 0, (1.7) u(x,s) = u 0 (x,s), 0 < x < 2π, −τ ≤ s ≤ 0. ( The effect of time delays in PDEs has been studied by different investigators to see whether or not such delays can destabilize the system under study [10,11,12,14,16,17,25,26,27,28].Oliva [28] is one of the investigators who considered dissipative scalar reactiondiffusion equations with boundary conditions which include small delays.He showed the global existence and uniqueness of solution in a convenient fractional power space.Also, Datko [10] and Datko et al. [11,12] studied certain hyperbolic partial differential equations with Neumann boundary conditions that include time delays.These equations are the Euler beam equation and the two-dimensional wave equation on a square.They showed that these equations could be destabilized when small delays are introduced into their boundary feedback controls.However, Friesecke [17] considered Hutchinson's equation which arises in population dynamics as a model for the evolution of a population with density distribution u.He studied equations of the following form: and showed that all nonnegative solutions of the initial boundary value problem stay bounded as t → ∞ in one, two, or more dimensions.Burgers equation with time delays was also investigated by Liu [25] who considered the following form of Burgers equation: with Dirichlet boundary condition.He showed that the delayed Burgers equation is exponentially stable if the delay parameter is sufficiently small.This paper is organized as follows.Section 2 analyzes the behavior of the solution of (1.1) without introducing a time delay.Section 3 discusses the behavior of the timedelayed Burgers equation (1.6).Section 4 presents some numerical results for both studies that support the analytical results, and we conclude in Section 5.

The generalized Burgers equation without a time delay
The forced Burgers equation without a time delay has been the subject of numerous studies [1,4,5,6,7,8,9,15,16,19,20,21,22,30,31].Ablowitz and De Lillo [1] considered Burgers equation where u = u(x,t) and G(x,t) is a given function.They linearized the initial value problem on the line for an integrable bounded function of time F(t) and discussed the asymptotic behavior of the solution for particular choices of F(t).Ito and Yan [20] studied the forced viscous scalar conservation law on (0,1) with the nonlinear flux feedback at the boundary They showed that under an appropriate growth condition on the flux function and nonlinear dissipation at the boundary, there exists an absorbing set that absorbs the whole space L ∞ (0,1), and they proved the existence of a compact global attractor in the L ∞topology.
Smaoui [31] studied the long-time dynamics of a system of reaction-diffusion equations that arise from the viscous forced Burgers equation where the force is sinusoidal, with periodic boundary conditions.He used a nonlinear transformation introduced by Kwak to embed the scalar Burgers equation into a system of reaction-diffusion equations.
He showed analytically as well as numerically that the two systems have a similar longtime dynamical behavior for large viscosity ν.

h(x)
= 0.In the following, we show that the generalized Burgers equation when h(x) = 0, and with periodic boundary condition, is globally stable.Let then Using integration by parts and the periodicity of u, we get Applying the Poincaré inequality and the zero-mean condition on u on the above equation, we obtain By Lyapunov theory, lim t→∞ u(x,t) = 0, which implies that (2.4) is globally asymptotically stable.
Global stability can also be shown by using control theory.Let u s be the steady state solution of u, then (2.9) If one defines the regulation error by then (2.4) becomes with periodic boundary control w(0,t) = f 0 , and w(2π,t) = f 1 , where f 0 and f 1 are scalar control inputs.Then by taking the control Lyapunov function and taking the time derivative of V , one can then find the control f 0 and f 1 that can enhance the negativity of (d/dt)(V ) which implies that w(x,t) = 0 or u(x,t) = u s .Thus, (2.4) is globally asymptotically stable in L 2 (0,2π).(For a complete stability analysis using control theory, the reader is referred to [3,9,19,23,24,26,27].)

h(x)
= 0.In this section, we show that in the Hilbert space H = H 2 (0,2π) consisting of 2π-periodic functions with zero mean, the first and second derivatives in L 2 (0,2π), and inner product u,v 2 = u xx v xx dx, the generalized Burgers equation has a unique steady state solution provided Proof.Suppose there are two steady state solution u and υ such that Multiplying the above equation by w, integrating from 0 to 2π, and using the periodicity of u and w lead to Again, using integration by parts on 2π 0 u(w 2 /2) x dx, we obtain Equation (2.22) can be written as it follows that then w = w x = 0, which implies u = υ.

The generalized Burgers equation with a time delay
The effect of time delays in certain partial differential equations has been the subject of recent studies [10,11,12,14,16,17,25,26,27,28].The question that is frequently asked is: can such delays destabilize a system which is stable in the absence of delays?Oliva [28] considered dissipative scalar reaction-diffusion equations that include the ones of the form subjected to boundary conditions that include small delays He proved the global existence and uniqueness of solutions in a convenient fractional power space.Furthermore, he showed that, for τ sufficiently small, all bounded solutions N. Smaoui and M. Mekkaoui 79 are asymptotic to the set of equilibria as t tends to infinity.On the other hand, Liu [25] considered the time-delayed Burgers equation with Dirichlet boundary condition.He showed that the delayed Burgers equation is exponentially stable if the delay parameter is sufficiently small.In this section, the generalized Burgers equation with time delay is studied: First, we show that the problem given by (3.4), (3.5), and (3.6) is well posed.We define the linear operator A by where H 2 (0,2π) consists of 2π-periodic functions with zero mean, two derivatives in L 2 (0,2π), and inner product u,v 2 = u xx v xx dx.It is well known that the operator A generates an analytic semigroup e At in L 2 (0,2π) (see Temam [34]).Also, we define the nonlinear operator where B is locally Lipschitz.If we denote then the generalized Burgers equation (3.4) can be written in terms of the above operators as Using Gronwall's inequality, we obtain (3.11) Lemma 3.1.The generalized Burgers equation ) Proof.See [28, Theorem 1].
In the following, we show that (3.12) does not blow up for finite time.Let nτ ≤ t ≤ (n + 1)τ (n = 0,1,...).First, we prove that for n = 0 and for any τ ≥ 0, (3.12) does not blow up for finite time.Then, we use continuation to show that this is true for all n.For n = 0 (i.e., 0 Using integration by parts on the right-hand side of (3.16) and making use of the periodicity of u, we get (3.17) Now using the fact that |u(t − τ)| ≤ u 0 C([−τ,0],H 1 (0,2π)) , for 0 ≤ t ≤ τ, and integrating by parts the last term of the right-hand side, then the above equation becomes N. Smaoui and M. Mekkaoui 81 Next, using Cauchy-Schwarz and Young's inequalities on the above equation, we get which implies that 2π 0 The same result can be shown for nτ ≤ t ≤ (n + 1)τ (n = 1,2,...) by applying the same procedure.Thus, for any τ > 0, the solution will not blow up in a finite time.
Before stating the main result about the exponential stability, the following notations are introduced.For a given initial condition u 0 = u 0 (x,s) ∈ C([−τ,0],H 1 (0,2π)), let where • denotes the L 2 -norm and u 0x 2 ) In (3.24), ω > 0 because 82 Generalized Burgers equation which is true only if Proof.Since then integrating by parts and using the periodicity of u, we get (3.44) Now using the Cauchy-Schwarz inequality, we obtain where ω is defined by (3.24).Solving this inequality, we obtain where ω is defined as in (3.24).

Numerical results
In this section, we would like to find a Fourier representation for the generalized Burgers equation without a time delay and the generalized Burgers equation with a time delay with periodic boundary conditions Equation (4.1) can be written as where Spectral approximation could be used to find the Fourier representation because of its accuracy and efficiency since we can expand the function u in terms of an infinite sequence of orthogonal functions But since most numerical methods based upon Fourier series cannot be implemented directly by standard treatment of Fourier series because the Fourier coefficients of an arbitrary complex-valued function are not known and must be approximated in some way, we use the discrete Fourier series [8].That is, for any integer N > 0, we consider the set of points The discrete Fourier coefficients of a complex-valued function u in [0,2π] with respect to these points are ũk All the nonlinear terms in (4.12) and (4.13) were evaluated in the physical space followed by the discrete Fourier transform to find the Fourier coefficients.The aliasing error was removed by truncation in the manner described in [8], that is, by performing all multiplication in a physical space followed by the discrete Fourier transform to determine the corresponding Fourier coefficients.Two computer programs that use a spectral Galerkin method with N = 256 were written to solve both (4.12) and (4.13).The value of N = 256 in those equations was chosen so that not only the truncation error is kept down to a minimum, but also the aliasing error caused by the nonlinear term is completely removed.The "slaved-frog" scheme was used [18].That is where u n = u(t n ), f n = f (t n ).This is obtained from the exact relation    decreasing), one can see that the solution decays to zero exponentially quickly when the delay τ is small (see Figure 4.4).The numerical results obtained are in accordance with the analytical ones presented in Sections 2 and 3.

Concluding remarks
In this paper, we studied the generalized Burgers equation with periodic boundary conditions on the interval [0,2π] with and without introducing a time delay for sufficiently large viscosity.By using Lyapunov theory, we showed that for the generalized Burgers equation without a time delay and when h(x) = 0, the equation is globally asymptotically stable.Moreover, we showed that when h(x) = 0, the steady state solution is bounded and unique.For the generalized Burgers equation with a time delay and when h(x) = 0, we showed that the equation is exponentially stable under small delays.We presented some numerical results by using the spectral method to support the analytical results given in Sections 2 and 3.The case when h(x) = 0 in the generalized time-delayed Burgers equation and the analysis of the behavior of its solution for different values of h(x) will be the subject of future studies.

Figure 4 .
Figure 4.1 presents the steady state solution of the generalized Burgers equation (4.1) without time delay when h(x) = 0.5sin(x) and u(x,0) = sin(x) for different viscosity.One

Figure 4 . 2 .
Figure 4.2.Steady state solutions of the generalized Burgers equation without time delay for different values of ν with h(x) = 0.5cos(x) and u(x,0) = cos(x).