Asymptotic Behavior And Existence Of Solution Of Semilinear Nonlocal Partial Differential Equation

In recent years, nonlocal equation has been applied verywell in many fields, such as anomalous diffusion, image processing, fluid mechanics, seismic analysis, soft matter research, viscoelastic damper, control and signal processing, etc. This paper is going to discuss the existence of semilinear nonlocal steady state solution of the equation and the nontrivial solutions and the corresponding asymptotic behavior of the solutions of reactiondiffusion equations with the method of variational. In this paper, problems will be discussed as follows:-Aαu = u- u3, x ∈(0, l)u |Dc = 0.D =(0, l), Dc= R1\D, the nonlocal operator Aαis defined as:Aαu =-D(Θ · D?u) =∫D∪Dτ(u(x)- u(y)) · γ(x, y)dy, 0 < α < 2.In the first chapter, It mainly introduces the research status quo of the partial differential equation, and the equation of the classification, including the main contents and conclusions of this paper.In the second chapter, It describes the nonlocal operator theory, including nonlocal operator, integral area, local integral operator, the kernel, equivalent space. In addition,some preliminary knowledge will also be contained in it.In the third chapter, it firstly converts the equation into solving the corresponding energy functional’s extrema problem in variational method, then uses weak compactness theorem and weak lower semicontinuous obtained the minimum solution will be found in the space of Hα2, which means weak solution exists in the space. Then according to the nature of the energy functional namely, if l >(2λ)1, then there is exists point u ∈ Hα2 such that I(u) < 0, that is to say the equation exists nontrivial solution. Finally, it considers the asymptotic behavior of solution,in the region(0, l),, if l < min(C-1α+1,(2λ)1α), arbitrary initial value u0(x) ∈ L2, then the semilinear nonlocal partial differential equation corresponding nonlocal reaction diffusion equation’s solution converge to the zero solution of the equation.In the fourth chapter, it illustrates some prospects of the topic, which including the definition of local operators in the unbounded region, the differences of the working space and the definition of bounded regional. It also discusses the difficulties exist in the asymptotic behavior for the nonlocal reaction diffusion equation.