Non-autonomous forms and Gaussian estimates

Abstract

Following the approach of J.-L. Lions we study non-autonomous Cauchy problems by form methods in this thesis. We replace the assumption of the closedness of the forms by their closability and extend the definition of weak solutions of the non-autonomous Cauchy problem to this more general situation. Then we prove the existence of solutions and their a priori properties. Uniqueness of solutions is shown in the closure of smooth functions. We prove that all solutions are in this closure for a class of weighted spaces with weak assumptions on the weight. The solutions are governed by strongly continuous evolution families if all solutions are in the closure of smooth functions. Under this assumption we study also the dual and final time problem as well as positivity of the evolution family. In a second part of this thesis we apply these existence result to non-autonomous, second-order differential forms with unbounded coefficients. Under some assumptions on the integrability of the coefficients and the geometry of the spaces we prove ultracontractivity of the evolution family. Moreover, we show upper bounds of the integral kernel using the perturbation method of E. B. Davies. These estimates are a generalization of the Gaussian estimates. Under stronger conditions on the second-order coefficients we obtain classical Gaussian estimates. Applying the upper bound to autonomous problems, we show that the bounds imply spectral independence and analyticity of the associated semigroup in an open sector.