Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient a is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative pr...

Existence and uniqueness results are given for secondorder parabolic and elliptic equations with variable coefficients in C domains in Sobolev spaces with weights allowing the derivatives of solutions to blow up near the boundary. The “number” of derivatives can be negative and fractional. The coefficients of parabolic equations are only assumed to be measurable in time.

Representation of functionals of non-Markov processes is studied for bounded and unbounded domains. These functionals are represented via solutions of backward parabolic Ito equations. This results is based on an analog of the second fundamental inequality and the related existence theorem are obtained for backward parabolic Ito equations. AMS 1991 subject classification: Primary 60J55, 60J60, ...

We establish the Harnack inequality for advection-diffusion equations with divergencefree drifts of low regularity. While our previous work [IKR] considered the elliptic case, here we treat the more challenging parabolic problem by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale-invariant spaces.

We analyze stability properties of BDF methods of order up to 5 for linear parabolic equations as well as of implicit–explicit BDF methods for nonlinear parabolic equations by energy techniques; time dependent norms play also a key role in the analysis.

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest ...

In this paper we establish some exact controllability results for systems of two parabolic equations. In a first part, we prove the existence of insensitizing controls for the L norm of the gradient of solutions of linear heat equations. Then, in the worst situation where null controllability for a system of two parabolic equations can hold, we prove this result for some general couplings. MSC:...

Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930...

This paper studies unique continuation for weakly degenerate parabolic equations in one space dimension. A new Carleman estimate of local type is obtained to deduce that all solutions that vanish on the degeneracy set, together with their conormal derivative, are identically equal to zero. An approximate controllability result for weakly degenerate parabolic equations under Dirichlet boundary c...

In this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality conditions in the form of an exact Pontryagin’s minimum principle for distributed and boundary controls (which can be unbounded) and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear parabolic equati...