Asymptotic Behavior Of Non-negative Solutions To Nonlinear Fractional Reaction-diffusion Equations

In this paper, we study the asymptotic behavior of non-negative nontrivial solution-s to the nonlinear fractional reaction-diffusion equation. For the initial value problems of nonlinear fractional non-autonomous reaction-diffusion equations with bounded and inte-grable initial conditions, whose diffusion coefficients depend on time variables, we analyse the large time behavior of non-negative nontrivial solutions, and discuss the conditions that make the mass functions defined by non-negative solutions of the systems tends to a positive number or zero respectively.In chapter two, for a class of nonlinear fractional non-autonomous reaction-diffusion equations dTu= -G(τ)(-△)α/2u-H(τ)F(u), whose diffusion coefficient G(τ) depends on time variable τ, and for a general nonlinear reaction function F(u), we give the asymp-totic behavior of non-negative nontrivial solutions of the systems, and discuss the conditions that make the mass functions M(τ)= ∫RN u(x, τ)dx defined by the non-negative solution keeps the positivity for all t> 0.In chapter three, for the Cauchy problem of a class of nonlinear fractional weak coupled reaction-diffusion equations ut= -(-△)α1/2u-vp, vt=-(-△)α2/2v-uq, the mass function M(t)=∫RN [u(x,t)+v(x, t)]dx defined by the non-negative nontrivial solutions(u(x, t), v(x, t)) of the systems decrease monotonically, we prove that the mass of systems is positive when p> 1+α2/N,q> 1+α1/N, and give the asymptotic relation when α1= α2. While when 1< p≤ 1+α2/N,1< q≤ 1+α1/N the mass of systems tends to zero as time increasing. Moreover, for the Cauchy problem of nonlinear fractional weak coupled reaction-diffusion equations, whose diffusion coefficients depend on time variable, we can obtain similar results by the method of this paper.