Lower bounds at infinity for solutions of differential equations with constant coefficients

Two extensions of a classical theorem of Rellich are proved: (1) LetP=P(−iϖ/ϖx) be a partial differential operator with constant coefficients in , let the manifolds contained in have codimension ≧k>0, and denote by Γ an open cone in intersecting each normal plane of every such manifold. If ,Pu=0 and it follows thatu=0. (2) Assume in addition that each irreducibe lfactor ofP van shes on a real hypersurface and that Γ contains both normal directions at some such point. If andP(D) u has compact support, the same condition withk=1 implies thatu has compact support. In both results the hypotheses on the cone Γ and on the operatorP are minimal.     

Decay at infinity of solutions to higher order partial differential equations: Removal of the curvature assumption

In an earlier paper a generalization of Rellich’s theorem on the Helmholz equation was obtained for a large class of higher order equationsP(1/i∂/∂x)u=f, subject to the condition that the Gaussian curvature ofP(ξ)=0 never vanish. This restriction is removed in this note.     

Lower bounds on the cost functional for systems governed by partial differential equations

Given two systems, one governed by a partial differential equation of a parabolic type and the other governed by a partial differential equation of a hyperbolic type, a cost functional is defined. For each system, an inequality is derived. From these inequalities, lower bounds on the cost functional can be estimated.     

On the behavior at infinity of solutions to differential equations in Hilbert spaces

The paper deals with the operator differential equation y(²ⁿ)(t)+Ay(ⁿ)(t)+By(t)=0, where A and B are self-adjoint commuting operators, in a Hilbert space. Criteria are established for stability and stabilization of solutions at infinity. In the case of growing solutions, their asymptotic behavior is investigated. Examples are given from mathematical physics. Bibliography: 21 titles. Dedicated to Olga Arsenievna Oleinik as a token of acknowledgment for her outstanding role in the development of the theory of differential equations Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000     

Gradient bounds for anisotropic partial differential equations

We consider solutions in the whole of the space of a partial differential equation driven by the anisotropic Laplacian. We prove a pointwise energy bound and we derive from that some rigidity results.     

bounds in some non-standard nonlinear problems in partial differential equations

A priori bounds are determined for certain energy expressions for a class of semi-linear parabolic and hyperbolic initial-boundary value problems when a combination of the values of the solution initially and at a later time is prescribed.     

On the behavior of solutions of operator-differential equations at infinity

For bounded solutions of an operator-differential equation y (t)=Ay (t) in a reflexive Banach space, we establish the existence of generalized limits (in Abel's sense) at infinity.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 809–813, July, 1994.This work was partially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite     

On differential properties of solutions of one class of pseudodifferential equations at infinity. I

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