Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families

Maximal Regularity for Evolution Equations Governed by Non-Autonomous Forms

We consider a non-autonomous evolutionary problem u̇(t) + A(t)u(t) = f(t), u(0) = u0 where the operator A(t) : V → V ′ is associated with a form a(t, ., .) : V × V → R and u0 ∈ V . Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space H such that V is continuously and densely embedded into H and given f ∈ L(0, T ;H) we are interested...

THE CAUCHY PROBLEM FOR p-EVOLUTION EQUATIONS

In this paper we deal with the Cauchy problem for evolution equations with real characteristics. We show that the problem is well-posed in Sobolev spaces assuming a suitable decay of the coefficients as the space variable x → ∞. In some cases, such a decay may also compensate a lack of regularity with respect to the time variable t.

Invariance of convex sets for non-autonomous evolution equations governed by forms

We consider a non-autonomous form a : [0, T ]× V × V → C where V is a Hilbert space which is densely and continuously embedded in another Hilbert space H . Denote by A(t) ∈ L(V, V ′) the associated operator. Given f ∈ L(0, T, V ′), one knows that for each u0 ∈ H there is a unique solution u ∈ H(0, T ;V ′) ∩ L(0, T ;V ) of u̇(t) +A(t)u(t) = f(t), u(0) = u0. This result by J. L. Lions is well-know...

Nonlocal Cauchy Problem for Nonautonomous Fractional Evolution Equations

During the past decades, the fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in engineering and physics; they attracted many researchers cf., e.g., 1–9 . On the other hand, the autonomous and nonautonomous evolution equations and related topics were studied in, for example, 6, 7, 10–20 , and the nonlocal Cauchy problem was considere...

Evolution Equations Governed by Elliptic Operators