Lower Bounds for Solutions of Parabolic Differential Inequalities

1967 ◽  
Vol 19 ◽  
pp. 667-672 ◽  
Author(s):  
Hajimu Ogawa
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Asymptotic Behaviour ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Lower Bounds ◽  
Second Order ◽  
Differential Inequalities ◽  
Linear Elliptic Operator ◽  
Image Position

Let P be the parabolic differential operatorwhere E is a linear elliptic operator of second order on D × [0, ∞), D being a bounded domain in Rn. The asymptotic behaviour of solutions u(x, t) of differential inequalities of the form1has been investigated by Protter (4). He found conditions on the functions ƒ and g under which solutions of (1), vanishing on the boundary of D and tending to zero with sufficient rapidity as t → ∞, vanish identically for all t ⩾ 0. Similar results have been found by Lees (1) for parabolic differential inequalities in Hilbert space.

Asymptotic Behaviour of Solutions of Parabolic Differential Inequalities

1962 ◽  
Vol 14 ◽  
pp. 626-631 ◽  
Author(s):  
Milton Lees
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Differential Inequality ◽  
Second Order ◽  
Differential Inequalities ◽  
Order Linear ◽  
Open Set ◽  
Asymptotic Behaviour Of Solutions ◽  
Image Position

Let there be given a parabolic differential operatorwhere A is a second order linear elliptic (<0) differential operator in an open set Ω ⊂ Rn, having coefficients depending on x ∈ Ω and t ∈ [0, ∞]. Recently, Protter (1) investigated the asymptotic behaviour of functions u(x, t) that satisfy the differential inequality(1.1)Under suitable restrictions on the functions ci(t) and the coefficients of A, he proved that any solution of (1.1), subject to certain homogeneous boundary conditions, that vanishes sufficiently fast, as t → ∞, must be identically zero in Ω × [0, ∞). For example, conditions are given under which no solution of (1.1) can vanish faster than e-λt, ∀ λ > 0, unless identically zero.

Half-eigenvalues of elliptic operators

2002 ◽  
Vol 132 (6) ◽  
pp. 1439-1451 ◽  
Author(s):  
Bryan P. Rynne
Keyword(s):  
Differential Operator ◽  
Bounded Domain ◽  
Trivial Solution ◽  
Partial Differential Operator ◽  
Elliptic Operators ◽  
Second Order ◽  
Partial Differential ◽  
Carathéodory Function ◽  
Image Position ◽  
Semilinear Problem

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.We also consider the semilinear problem where f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω, and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

On nonlinear perturbations of linear second order elliptic boundary value problems

1978 ◽  
Vol 84 (1) ◽  
pp. 143-157 ◽  
Author(s):  
P. M. Fitzpatrick
Keyword(s):  
Asymptotic Behaviour ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Nonlinear Perturbations ◽  
Elliptic Boundary Value ◽  
Image Position

AbstractLet Ω ⊆ n be open and bounded, with ∂Ω smooth. Sufficient conditions on f are given in order that for p > n, the equationhas a solution for every h ∈ Lp; L is a second order symmetric elliptic operator and B represents either the first, second or third boundary value problems. These conditions are in terms of the asymptotic behaviour of f, in its second variable, in relation to those λ for which there is a nontrivial solution of

8.—On Second-order Differential Inequalities

1974 ◽  
Vol 72 (2) ◽  
pp. 109-127 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Second Order ◽  
Differential Inequalities ◽  
Existence Of Positive Solutions ◽  
Image Position

SynopsisThis paper is devoted to a study of differential equations and inequalities of the formandThe results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations

2015 ◽  
Vol 145 (5) ◽  
pp. 1007-1028 ◽  
Author(s):  
Jaroslav Jaroš ◽  
Kusano Takaŝi
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Nonlinear Differential Equations ◽  
Radial Symmetry ◽  
Second Order ◽  
Accurate Information ◽  
Regularly Varying Functions ◽  
Cyclic System ◽  
Regularly Varying ◽  
Image Position

The n-dimensional cyclic system of second-order nonlinear differential equationsis analysed in the framework of regular variation. Under the assumption that αi and βi are positive constants such that α1 … αn > β1 … βn and pi and qi are regularly varying functions, it is shown that the situation in which the system possesses decreasing regularly varying solutions of negative indices can be completely characterized, and moreover that the asymptotic behaviour of such solutions is governed by a unique formula describing their order of decay precisely. Examples are presented to demonstrate that the main results for the system can be applied effectively to some classes of partial differential equations with radial symmetry to provide new accurate information about the existence and the asymptotic behaviour of their radial positive strongly decreasing solutions.

scholarly journals Positive functionals and oscillation criteria for second order differential systems

1979 ◽  
Vol 22 (3) ◽  
pp. 277-290 ◽  
Author(s):  
Garret J. Etgen ◽  
Roger T. Lewis
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Second Order ◽  
Linear Operators ◽  
The Self ◽  
Differential Systems ◽  
Bounded Linear ◽  
Positive Functionals ◽  
Image Position ◽  
Adjoint Operators

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation

Lp-Theory of degenerate-elliptic and parabolic operators of second order

1983 ◽  
Vol 95 (1-2) ◽  
pp. 95-113 ◽  
Author(s):  
Baoswan Wong-Dzung
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Positive Semidefinite ◽  
Second Order ◽  
Minimal Realization ◽  
Elliptic Differential Operator ◽  
Second Derivatives ◽  
Degenerate Elliptic ◽  
Image Position ◽  
Lp Theory

SynopsisWe consider the formal operator given byin the Banach space X = LP(Rn), 1<p<∞. The coefficients ajk(x), aj(x), and a(x) are real-valued functions, ajk ε C2(Rn) has bounded second derivatives, aj ε Cl(Rn) has bounded first derivatives, and aεL∞(Rn). Furthermore, we assume that the n × n matrix (ajk(x)) is symmetric and positive semidefinite (i.e. ajk(x)ξjξk≧0 for all (ξ1,…,ξn)ε Rn and x ε Rn). We prove that the degenerate-elliptic differential operator given by –A and restricted to , the minimal realization of –A, is essentially quasi-m-dispersive in Lp(Rn), (hence that the minimal realization of +A is quasi-m-accretive) and that its closure coincides with the maximal realization of –A.

Oscillatory and asymptotic properties of a class of operator-differential inequalities

1984 ◽  
Vol 96 (1-2) ◽  
pp. 5-13 ◽  
Author(s):  
A. D. Myshkis ◽  
D. D. Bainov ◽  
A. I. Zahariev
Keyword(s):  
Asymptotic Behaviour ◽  
Asymptotic Properties ◽  
Neutral Type ◽  
Differential Inequalities ◽  
Functional Differential ◽  
Image Position ◽  
Oscillatory Properties

SynopsisThe present paper studies some asymptotic (including oscillatory) properties of the solutions of operator-differential inequalities of the formwhere(the latter symbol denotes the space of locally summable functions).As an application of the results obtained, theorems are proved for the asymptotic behaviour of the solutions of certain classes of functional-differential and integro-differential neutral-type equations.

Discrete spectra criteria for singular differential operators with middle terms

1975 ◽  
Vol 77 (2) ◽  
pp. 337-347 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Continuous Functions ◽  
Adjoint Extension ◽  
Image Position ◽  
Discrete Spectra ◽  
Bounded Below

Let l be the differential operator of order 2n defined bywhere the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

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