The principal eigenvalue of a space–time periodic parabolic operator

• DOI: 10.1007/s10231-008-0075-4
• Publication Date: 2008-03-31T08:57:19Z
• Authors: Gregoire Nadin
• Keywords: Nabla symbol; Operator (biology)

This paper deals with the generalized principal eigenvalue of the parabolic operator $${\mathcal{L}\phi = \partial_{t}\phi - \nabla \cdot(A(t, x)\nabla\phi) + q(t, x) \cdot \nabla\phi - \mu(t, x)\phi}$$ , 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.