Existence and uniqueness of the solution of a space–time periodic reaction–diffusion equation

This paper is concerned with the study of the periodic solutions and the entire solutions of the equation:equation(1)∂tu−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u)∂tu−∇⋅(A(t,x)∇u)+q(t,x)⋅∇u=f(t,x,u) where the diffusion matrix A, the advection term q and the reaction term f are periodic in t and x. We prove that the sign of the periodic principal eigenvalue associated with the linearized problem determines the existence and the uniqueness of the periodic solution. Introducing another eigenvalue, we are able to state uniqueness conditions for the entire solution and to derive the asymptotic behavior of the solutions of the associated Cauchy problem.