Nondivergence Parabolic Equations Research Articles

On the Stabilization Rate of Solutions of the Cauchy Problem for Nondivergent Parabolic Equations with Growing Lower-Order Terms

On the Stabilization Rate of Solutions of the Cauchy Problem for Nondivergent Parabolic Equations with Growing Lower-Order Terms

On the Harnack inequality for non-divergence parabolic equations

In this paper we propose an elementary proof of the Harnack inequality for linear parabolic equations in non-divergence form.

Regularity Estimates for Nondivergence Parabolic Equations on Generalized Orlicz Spaces

Abstract In this paper, by using a new approach, we prove regularity estimates for the solution to the nondivergence parabolic equation on generalized Orlicz spaces. Our approach can be viewed as a combination of representation theorems in partial differential equations and harmonic analysis techniques.

The Inverse Problem of Simultaneous Determination of the Two Time-Dependent Lower Coefficients in a Nondivergent Parabolic Equation in the Plane

We prove existence and uniqueness theorems for the solution of the inverse problem of simultaneous determination of the t-dependent coefficients of u and ux in a nondivergent parabolic equation with two independent variables from integral observation of x. Estimates of the maxima of the moduli of these coefficients with constants explicitly expressed in terms of the input data of the problem are given. An example of an inverse problem to which the proved theorems apply is presented.

<inline-formula><tex-math id="M1">\begin{document}$ L^{p(\cdot)} $\end{document}</tex-math></inline-formula>-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients

We prove the interior $ L^{p(\cdot)} $-estimates for the Hessian of strong solutions to nondivergence parabolic equations $ u_{t}(x,t)-a_{ij}(x,t)D_{ij}u(x,t) = f(x,t) $ and elliptic equations $ a_{ij}(x)D_{ij}u(x) = f(x) $, respectively. Besides a natural assumption that $ p(\cdot) $ is $ \log $-Hölder continuous, we also assume that the coefficients $ a_{ij}(x,t) $ and $ a_{ij}(x) $ are merely measurable in one of spatial variables and have small BMO semi-norms with respect to other variables.

Hessian Estimates for Nondivergence Parabolic and Elliptic Equations with Partially BMO Coefficients

In this paper, we mainly prove a local Calderon–Zygmund estimate in the Lorentz spaces with a variable power $$p(\cdot )$$ to the Hessian of nondivergence parabolic equations $$u_{t}(x,t)-a_{ij}(x,t)D_{ij}u(x,t)=f(x,t)$$, under assumptions that the variable exponent $$p(\cdot )$$ is $$\log $$-Holder continuous, and the coefficient is a small partially BMO matrix which means that $$a_{ij}(x,t)$$ is merely measurable in one of spatial variables and have small BMO semi-norms with respect to other variables. In addition, we also derive a similar result for nondivergence elliptic equations $$a_{ij}(x)D_{ij}u(x)=f(x)$$ with small partially BMO coefficients.

The Inverse Problem of Simultaneous Determination of the Two Lower Space-Dependent Coefficients in a Parabolic Equation

Existence and uniqueness theorems for the solution of the inverse problem of simultaneous determination of x-dependent coefficients at u and ux in a nondivergent parabolic equation from integral observation are obtained. Estimates of the maximum absolute values of these coefficients with constants explicitly expressed via the input data of the problem are given. An example of an inverse problem to which the proved theorems apply is presented.

On the Stabilization Rate of Solutions of the Cauchy Problem for Nondivergent Parabolic Equations with Growing Lower-Order Term

In the Cauchy problem L1u≡Lu+(b,∇u)+cu-ut=0,(x,t)∈D,u(x,0)=u0(x),x∈RN, for nondivergent parabolic equation with growing lower-order term in the half-space D=RN×[0,∞), N⩾3, we prove sufficient conditions for exponential stabilization rate of solution as t→+∞ uniformly with respect to x on any compact K in RN with any bounded and continuous in RN initial function u0(x).

Critical extinction exponent for a doubly degenerate non-divergent parabolic equation with a gradient source

We concern with the extinction phenomenon of a non-divergent parabolic equation with a nonlinear gradient source. The critical extinction exponent of the solution is obtained.

Non-divergence parabolic equations of second order with critical drift in Lebesgue spaces

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift−ut+Lu=−ut+∑ijaijDiju+∑biDiu=0(≥0,≤0) in some domain Q⊂Rn+1. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift b is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is a “universal” spectral gap for the associated elliptic operator. The counterpart for uniformly elliptic equations of second order in non-divergence form is shown in [19].

Global weighted estimates in Orlicz spaces for second-order nondivergence parabolic equations

In this paper we prove the following global weighted estimates in Orlicz spaces f∈Lwϕ(ΩT)⇒u,Du,D2u,ut∈Lwϕ(ΩT) for second-order parabolic equations of nondivergence form with small BMO coefficients {ut−aij(x,t)uxixj=fin ΩT,u(x,t)=0on ∂pΩT.

Oblique derivative problem for non-divergence parabolic equations with time-discontinuous coefficients in a wedge

We consider an oblique derivative problem in a wedge for non-divergence parabolic equations with time-discontinuous coefficients. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces.

Weighted estimates for nondivergence parabolic equations in Orlicz spaces

We obtain the global weighted Orlicz regularity of the maximum order derivatives of the solution to the Dirichlet problem for nondivergence parabolic equations in a bounded domain. We find correct conditions on the associated weight and Young function. The coefficients are assumed to have small bounded mean oscillation (BMO) seminorms.

Mixed Boundary-Value Problem for Linear Second-Order Nondivergent Parabolic Equations with Discontinuous Coefficients

The mixed boundary-value problem is considered for linear second-order nondivergent parabolic equations with discontinuous coefficients satisfying the Cordes conditions. The one-valued strong (almost everywhere) solvability of this problem is proved in the space Ŵ 2,1 , where p belongs to the same segment containing point 2.

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.

Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero

Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero

WEIGHTED SOLVABILITY FOR PARABOLIC EQUATIONS WITH PARTIALLY BMO COEFFICIENTS AND ITS APPLICATIONS

AbstractWe consider the weighted$L_p$solvability for divergence and nondivergence form parabolic equations with partially bounded mean oscillation (BMO) coefficients and certain positive potentials. As an application, global regularity in Morrey spaces for divergence form parabolic operators with partially BMO coefficients on a bounded domain is established.

Necessary and sufficient condition for the stabilization of the solution to the initial-boundary value problem for second-order nondivergence parabolic equations

Necessary and sufficient condition for the stabilization of the solution to the initial-boundary value problem for second-order nondivergence parabolic equations

Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabr\'e proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabr\'e's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.

Necessary and sufficient conditions for the stabilization of a solution to the Dirichlet problem for parabolic equation

We obtain necessary and sufficient conditions for the stabilization to zero of the solution to the Dirichlet problem for a nondivergence parabolic equation with measurable coefficients. The conditions are formulated in terms of the divergence of Wiener series containing the heat capacity introduced by E. M. Landis. In the case of the heat equation, these conditions yield the stabilization criterion. Bibliography: 3 titles.