Abstract
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.
Highlights
Non-autonomous evolution equations are partial differential equations in which relevant coefficients of the differential operator and/or in the boundary conditions are time-dependent, allowing for underlying models that are variable over time
If the coefficients of a non-autonomous equation are piecewise constant, one may find a solution by following the orbit of the semigroup governing a given problem as long as the coefficients stay constant; “freeze” the system; use the final state as an initial condition for a new evolution equation with new coefficients, and so on: this boils down to consider the composition of a finite numbers of semigroups
Lions shows that well-posedness in Hilbert space can be proved under much weaker assumptions, most notably mere measurability of the time dependence, provided the problem has a nice variational structure: this is typically the case if the differential equation is parabolic at any given time
Summary
Non-autonomous evolution equations are partial differential equations in which relevant coefficients of the differential operator and/or in the boundary conditions are time-dependent, allowing for underlying models that are variable over time. Several applications are reviewed in Section 6: we discuss well-posedness and qualitative properties of dynamical systems on undirected graphs tightly related to the theory of dynamic (positive) graphs discussed in [24] as well as models of Black–Scholes-types equations with time-dependent volatility [25]; we extend the kernel estimates in [26] to more general non-autonomous diffusion equations on possibly infinite networks; and we prove Gaussian bounds for the heat kernel for a large class of elliptic operators with time-dependent, possibly complex coefficients, deducing the main results in [15, 17] as special cases
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