The principal eigenvalue of a space-time periodic parabolic operator

This paper deals with the generalized principal eigenvalue of the parabolic operator L phi = partial derivative(t)phi - del center dot (A(t, x)del phi) + q(t, x) center dot del phi - mu(t, x)phi, where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.