Long Time Behavior Of Implicit Euler Method For The Space-Fractional Allen-Cahn Equation

At present,fractional differential equations have been widely used in physics,molec-ular dynamics,engineering,biology,economics and other fields.The diffusion of normal particles is essentially a markov localized motion,while anomalous diffusion is actually a non-markov local motion.To describe this nonlocal diffusion phenomenon,we need to replace the integer order reaction diffusion equation with the fractional order reac-tion diffusion equation.Therefore,there are more and more researches on the nonlocal(space-fractional)properties of the Allen-Cahn equation.The space-fractional Allen-Cahn equation with initial boundary value condition,whose fractional Laplacian operator is defined by fourier transform,is considered in this paper.First,in the continuous case,a prior estimates for the solution of the equation in L2?Hpae??Lq spaces are given.And we conclude that the equation is dissipative in the L2 space.Then,to obtain its time semi-discrete scheme,the implicit Euler method is used to discrete the space-fractional Allen-Cahn equation.Same as continuous,we also give some a prior estimates of the solution in L2?Hpaer??Lq spaces under the semi-discretization scheme,conclude that the discrete system is dissipative in the L2 space,and derive estimates for the difference of solutions in the semi-discretization scheme.Finally,the finite time uniform error estimate is given.The main results for semi-discrete systems are as follows:1.If the time step ? satisfies certain conditions respectively,and ?u0? ?R,then,in L2 space,we have?un???'0,n??2t0(R);in Hper? space,we have?un????1,n??2to(R)+2r;in Lq space,we have?un?L?i??i,n??ti(R).where ?'0,?1,?'i are all constants.2.If u0=u0 and ?u0???R1 we can obtain the results about finite time uniform error estimate(?)where the constant C(T,R1)is independent of ?.