Some spectral properties of differential operators

Mathematical Notes -     

Some spectral properties ofm-accretive differential operators

Form-accretive operatorT=T ₀+q compactness of its resolventR(z, T), zP(T) and ofR(z, T)–R(z, T ₀),zP(T)P(T ₀) under suitable assumptions onT ₀ and their perturbationq is established. This result is used in the study of spectral properties ofT andT ₀.     

Some properties of third order differential operators

Consider the third order differential operator L given by and the related linear differential equation L(x)(t) + x(t) = 0. We study the relations between L, its adjoint operator, the canonical representation of L, the operator obtained by a cyclic permutation of coefficients a _i , i = 1,2,3, in L and the relations between the corresponding equations.We give the commutative diagrams for such equations and show some applications (oscillation, property A).     

On nonnegative solutions of elliptic and parabolic equations

A theorem on the nonexistence of a nonnegative nontrivial generalized solution inR ⁿ is proved for general quasilinear second-order degenerate elliptic equations. Analogous results are obtained for a large class of systems of partial differential equations, second-order parabolic and inverse parabolic equations, which are nonlinear and may be degenerate.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 114–136, 1992.     

Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems

Summary Monotonicity methods are developped to investigate attractivity properties of non-negative stationary solutions for a class of nonlinear degenerate parabolic problems in any space dimension. Applications to specific problems suggested by population dynamics are also discussed.     

Stabilization of solutions to Shilov-type parabolic systems with nonnegative genus

We examine some class of Shilov-type parabolic systems with a nonnegative genus and bounded smooth coefficients depending on time and spatial variables and study the behavior of solutions when t tends to infinity. The initial data at t = 0 belong to a wide class of distributions.     

Existence of nonnegative solutions for parabolic problem on Dirichlet forms

In this work, we investigate perturbed parabolic problem on measure spaces. For strongly local Dirichlet form, we obtain some results on existence and uniqueness of nonnegative solution. We also presented some examples as an applications.     

Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators

In this paper we consider initial boundary value problem for semilinear parabolic equations involving strongly degenerate elliptic differential operators. Depending on the concrete types of nonlinearity we establish the existence of compact connected global attractors of semigroups generated by the problem under consideration.     

Equivariant first order differential operators for parabolic geometries

Let G be a real semisimple Lie group, P a parabolic subgroup, V and W irreducible representations of P, G × p V and G × p W the associated homogeneous vector bundles. The G-equivariant first order differential operators from the first to the second bundle are determined and described using methods of Lie theory.     

On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics

Quasilinear parabolic functional differential equations containing multiple transformations of spatial variables are considered with the Neumann boundary-value conditions. Sufficient conditions of the Andronov-Hopf bifurcation of periodic solutions are obtained along with expansions of the solutions in powers of a small parameter. Spectral properties of the linearized elliptic operator of this problem are investigated. Necessary and sufficient conditions of normality are obtained for such operators. Examples illustrating their properties are given. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 21, Proceedings of the Seminar on Differential and Functional Differential Equations Supervised by A. Skubachevskii (Peoples’ Friendship University of Russia), 2007.     

Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries

We consider nonnegative solutions of a parabolic equation in a cylinder D×(0,T), where D is a noncompact domain of a Riemannian manifold. Under the assumption [IU] (i.e., the associated heat kernel is intrinsically ultracontractive), we establish an integral representation theorem: any nonnegative solution is represented uniquely by an integral on (D×{0})∪(M_∂D×[0,T)), where M_∂D is the Martin boundary of D for the associated elliptic operator. We apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions. We also show that [IU] implies the condition [SP] (i.e., the constant function 1 is a small perturbation of the elliptic operator on D).     

Some properties of solutions for a fourth‐order parabolic equation

This paper is concerned with a fourth‐order parabolic equation in one spatial dimension. On the basis of Leray–Schauder's fixed point theorem, we prove the existence and uniqueness of global weak solutions. Moreover, we also consider the regularity of solution and the existence of global attractor. Copyright © 2012 John Wiley & Sons, Ltd.