The Dirichlet problem for non‐divergence parabolic equations with discontinuous in time coefficients in a wedge

We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted ‐spaces. The concept of “critical exponent” introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi‐linear non‐divergence parabolic equations with discontinuous in time coefficients in cylinders , where Ω is a bounded domain with an edge or with a conical point.     

The Dirichlet problem for second order parabolic operators in non‐cylindrical domains

In this paper we develope a perturbation theory for second order parabolic operators in non‐divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with L^p ‐data on the parabolic boundary (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the discontinuous second‐order deviated Dirichlet problem with non‐monotone conditions

We provide a new result on the existence of extremal solutions for second‐order Dirichlet problems with deviation argument. As a novelty in this work, the nonlinearity need not be continuous or monotone. In order to obtain this new result, we use a generalized monotone method coupled with lower and upper solutions.     

Solvability of parabolic equations in divergence form with partially BMO coefficients

We prove the solvability of second order parabolic equations in divergence form with leading coefficients aij measurable in (t,x¹) and having small BMO (bounded mean oscillation) semi-norms in the other variables. Additionally we assume a¹¹ is measurable in x¹ and has small BMO semi-norms in the other variables. The corresponding results for the Cauchy problem are also established. Parabolic equations in Sobolev spaces with mixed norms are also considered under the same conditions of the coefficients.     

Uniqueness for second-order parabolic equations with discontinuous coefficients

We prove uniqueness of the good solution to the Cauchy–Dirichlet (C–D) problem for linear non-variational parabolic equations with the coefficients of the principal part with discountinuities, in cases in which in general uniqueness of strong solutions in Sobolev spaces does not hold. In particular, we prove uniqueness when the discontinuities of the coefficients are contained in a hyperplane t = t ₀ and, with an extra condition on the eigenvalues of the matrix, in a line segment x = x ₀. Mathematics Subject Classification. 35A05, 35K10, 35K20 Dedicated to the memory of Gene Fabes.     

Boundary value problem for a global‐in‐time parabolic equation

The aim of this paper is to draw attention to an interesting semilinear parabolic equation that arose when describing the chaotic dynamics of a polymer molecule in a liquid. This equation is nonlocal in time and contains a term, called the interaction potential, that depends on the time‐integral of the solution over the entire interval of solving the problem. In fact, one needs to know the “future” in order to determine the coefficient in this term, that is, the causality principle is violated. The existence of a weak solution of the initial boundary value problem is proven. The interaction potential satisfies fairly general conditions and can have arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval.     

Gradient estimates for parabolic equations in generalized weighted Morrey spaces

We consider the Cauchy–Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain.The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others.We obtain Calderón–Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.     

Long‐time stability of finite element approximations for parabolic equations with memory

In this article, we derive the sharp long‐time stability and error estimates of finite element approximations for parabolic integro‐differential equations. First, the exponential decay of the solution as t → ∞ is studied, and then the semidiscrete and fully discrete approximations are considered using the Ritz‐Volterra projection. Other related problems are studied as well. The main feature of our analysis is that the results are valid for both smooth and nonsmooth (weakly singular) kernels. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 333–354, 1999     

Nonlinear anisotropic parabolic equations in with locally integrable data

In this paper, we prove existence and regularity of weak solutions for a class of nonlinear anisotropic parabolic problems in with locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. Copyright © 2012 John Wiley & Sons, Ltd.     

Second‐order elliptic and parabolic equations with BMO coefficients in the whole space

In this paper, we obtain the global regularity estimates in Orlicz spaces for second‐order divergence elliptic and parabolic equations with BMO coefficients in the whole space. In fact, the global result can follow from the local estimates. As a corollary we obtain L^p‐type regularity estimates for such equations. Copyright © 2011 John Wiley & Sons, Ltd.     

The Dirichlet problem for Hessian quotient equations in exterior domains

In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian quotient equations in exterior domains.     

On the decay of the solutions of second order parabolic equations with Dirichlet conditions

We use rearrangement techniques to investigate the decay of the parabolic Dirichlet problem in a bounded domain. The coefficients of the second order term are used to introduce an isoperimetric problem. The resulting isoperimetric function together with the divergence of the first order coefficients and the value distribution of the zero order part are then used to construct a symmetric comparison equation having a slower heat‐flow than the original equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Cauchy problem for second order strictly hyperbolic equations with non–regular coefficients

In this paper we shall consider some necessary and sufficient conditions for well–posedness of second order hyperbolic equations with non–regular coefficients with respect to time. We will derive some optimal regularities for well–posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory.     

Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations

A Discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi‐discrete and a family of fully‐discrete time approximate schemes are formulated. These schemes are symmetric. Hp‐version error estimates are analyzed for these schemes. For the semi‐discrete time scheme a priori L^∞(H¹) error estimate is derived and similarly, l^∞(H¹) and l²(H¹) for the fully‐discrete time schemes. These results indicate that spatial rates in H¹ and time truncation errors in L² are optimal. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Well‐posedness and regularity results for a 2m‐th order parabolic equation in symmetric conical domains of ℝN+1

In this paper, we use the domain decomposition method to prove well‐posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high‐order parabolic equation set in conical time‐dependent domains of . Copyright © 2017 John Wiley & Sons, Ltd.     

One‐step boundary knot method for discontinuous coefficient elliptic equations with interface jump conditions

This study makes the first attempt to apply the boundary knot method (BKM), a meshless collocation method, to the solution of linear elliptic problems with discontinuous coefficients, also known as the elliptic interface problems. The additional jump conditions are usually required to be prescribed at the interface which is used to maintain the well‐posedness of the considered problem. To solve the problem efficiently, the original governing equation is first transformed into an equivalent inhomogeneous modified Helmholtz equation in the present numerical formulation. Then the computational domain is divided into several subdomains, and the solution on each subdomain is approximated using the BKM approach. Unlike the conventional two‐step BKM, this study presents a one‐step BKM formulation which possesses merely one influence matrix for inhomogeneous problems. Several benchmark examples with various discontinuous coefficients have been tested to demonstrate the accuracy and efficiency of the present BKM scheme. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1509–1534, 2016     

Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse

The present paper considers the Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse. No restrictions on the multiplicities of the roots of the characteristic polynomial are assumed.