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We develop a variational approach in order to study qualitative properties of nonautonomous parabolic equations. Based on the method of product integrals, we discuss invariance properties and ultracontractivity of evolution families in Hilbert space. Our main results give sufficient conditions for the heat kernel of the evolution family to satisfy Gaussian‐type bounds. Along the way, we study examples of nonautonomous equations on graphs, metric graphs, and domains.

Ultracontractivity and Gaussian bounds for evolution families associated with nonautonomous forms