Maximal space regularity for abstract linear non-autonomous parabolic equations

On maximal parabolic regularity for non-autonomous parabolic operators

We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r 6= 2. This allows us to prove maximal parabolic L r-regularity for discontinuous non-autonomous second-order divergence form ...

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...

Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations

In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and establish smoothness properties of solutions. The results are based on maximal regularity estimates in continuous interpolation spaces. An important new ingredient ...

Mapping properties of heat kernels, maximal regularity, and semi-linear parabolic equations on noncompact manifolds

Let L : C∞(M ;E)→ C∞(M ;E) be a second order, uniformly elliptic, semipositive-definite differential operator on a complete Riemannian manifold of bounded geometry M , acting between sections of a vector bundle with bounded geometry E over M . We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form f...

Maximal Regularity for Evolution Equations