Stability Analysis For Nonlocal Dispersal Equations

Nonlocal operators can describe the process of population movement in space more accurately than Laplace operators.So more and more nonlocal diffusion models are used in the study of biological population diffusion.Stability analysis has important practical value and theoretical significance in the study of differ-ential equations,The stability analysis of nonlocal diffusion model has also been widely concernedIn this paper,we consider the following special nonlocal dispersal equation ut=J*up-up+f(x,u),in the homogeneous Dirichlet condition.Where J is a nonnegative kernel function,p is a constant no more than 1.The main results are divided into the following two partsThe equation is a general nonlocal diffusion equation when p=1.We estab-lished a special form of the maximum principle,and analyzed the stability of the equilibrium solution by using the super and subsolution theorem.The conditions for the existence of solutions in finite time or long time are givenThe equation becomes nonlinear nonlocal diffusion equation when p>1 The nonlinear property is not only reflected in the reaction term,but also in the diffusion term.We found that the comparison principle is also true for the positive solution of the equation.When the reaction term is a linear function of u,the existence and uniqueness of the solution is also true.Then,we proved the upper and lower solution principle for the positive solution and analyzed the stability of the equilibrium solution.Finally,we given the conditions of asymptotic stability and instability of equilibrium solution and studied the blow up behavior of solution as f(x,u)=?up.