scholarly journals Uniqueness of semilinear elliptic inverse problem

2003 ◽  
Vol 2003 (67) ◽  
pp. 4217-4227
Author(s):  
Chaochun Qu ◽  
Ping Wang
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Unique Solution ◽  
Dirichlet Condition ◽  
Elliptic Differential Equation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Semilinear Elliptic ◽  
Necessary And Sufficient

We consider the uniqueness of the inverse problem for a semilinear elliptic differential equation with Dirichlet condition. The necessary and sufficient condition of unique solution is obtained. We improved the results obtained by Isakov and Sylvester (1994) for the same problem.

scholarly journals A blow-up dichotomy for semilinear fractional heat equations

2020 ◽  
Author(s):  
Robert Laister ◽  
Mikołaj Sierżęga
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Heat Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Whole Space ◽  
Necessary And Sufficient

Abstract We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.

scholarly journals Symmetry and concentration behavior of ground state in axially symmetric domains

2004 ◽  
Vol 2004 (12) ◽  
pp. 1019-1030
Author(s):  
Tsung-Fang Wu
Keyword(s):  
Ground State ◽  
Semilinear Elliptic Equation ◽  
Bounded Domains ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Axially Symmetric ◽  
Ground State Solutions ◽  
Semilinear Elliptic ◽  
Necessary And Sufficient ◽  
Concentration Behavior

We letΩ(r)be the axially symmetric bounded domains which satisfy some suitable conditions, then the ground-state solutions of the semilinear elliptic equation inΩ(r)are nonaxially symmetric and concentrative on one side. Furthermore, we prove the necessary and sufficient condition for the symmetry of ground-state solutions.

scholarly journals ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS

2014 ◽  
Vol 57 (3) ◽  
pp. 543-554
Author(s):  
JANNE HEITTOKANGAS ◽  
ATTE REIJONEN
Keyword(s):  
Differential Equation ◽  
Bergman Space ◽  
Nontrivial Solution ◽  
Unit Disc ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Zeros ◽  
Zero Sequence ◽  
Zeros Of Solutions ◽  
Necessary And Sufficient

AbstractIf A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

scholarly journals Lyapunov-Krasovskii stability analysis of nonlinear integro-differential equation

2018 ◽  
Vol 7 (2) ◽  
pp. 53
Author(s):  
Prebo Jackreece
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Integro Differential Equation ◽  
Necessary And Sufficient

The purpose of this paper is to develop a qualitative stability analysis of a class of nonlinear integro-differential equation within the framework of Lyapunov-Krasovskii. We show that the existence of a Lyapunov-Krasovskii functional is a necessary and sufficient condition for the uniform asymptotic stability of the nonlinear Volterra integro-differential equations.

scholarly journals Existence of Ψ Bounded Solutions for a Lyapunov Matrix Dierential Equation

2014 ◽  
Vol 52 (2) ◽  
Author(s):  
Aurel Diamandescu
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Bounded Solution ◽  
Sufficient Condition ◽  
Matrix Differential Equation ◽  
Bounded Solutions ◽  
Necessary And Sufficient Condition ◽  
Lyapunov Matrix ◽  
Matrix Differential ◽  
Necessary And Sufficient

AbstractIt is proved a necessary and sufficient condition for the existence of at least one Ψ- bounded solution of a linear non- homogeneous Lyapunov matrix differential equation. In addition, it is given a result in connection with the asymptotic behavior of the Ψ- bounded solutions of this equation.

scholarly journals A necessary and sufficient condition for exponentially bounded existence and uniqueness

1973 ◽  
Vol 8 (1) ◽  
pp. 133-135 ◽  
Author(s):  
David Lowell Lovelady
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
Necessary And Sufficient

A condition which was previously found to be sufficient for global existence and uniqueness of solutions of an ordinary differential equation is shown herein to be necessary, if it is also required that solutions are exponentially bounded.

scholarly journals A necessary and sufficient condition for the existence of a mobile singular points for a third-order nonlinear differential equation

2021 ◽  
pp. 49-55
Author(s):  
Темирхан Султанович Алероев ◽  
Магомедюсуф Владимирович Гасанов
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Order Equation ◽  
Singular Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Degree Polynomial ◽  
Third Order ◽  
The Right ◽  
Necessary And Sufficient

Рассматривается нелинейное уравнение третьего порядка с полиномом второй степени в правой части. Отличительной чертой этого класса уравнений является наличие подвижных особенностей, что делает эти уравнения неразрешимыми в квадратурах. В работе получены интервальные критерии существование подвижных особых точек. Представленная теория является подспорьем для написания различных алгоритмов в различных программных комплексах для нахождения подвижных особых точек. A nonlinear third-order equation with second degree polynomial on the right. The hallmark of this class equations is the presence of movable singularities, which makes these equations undecidable in quadratures. The work obtained interval criteria the existence of movable singular points. The theory presented is help for writing various algorithms in various software complexes for finding movable singular points.

scholarly journals A necessary and sufficient condition for the existence of a unique solution of a discrete boundary value problem

2003 ◽  
Vol 2003 (39) ◽  
pp. 2455-2463 ◽  
Author(s):  
Raghib Abu-Saris ◽  
Wajdi Ahmad
Keyword(s):  
Boundary Value Problem ◽  
Unique Solution ◽  
Boundary Value ◽  
Linear Difference Equation ◽  
Fundamental Question ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Constant Coefficients ◽  
Necessary And Sufficient

Akth-order linear difference equation with constant coefficients subject to boundary conditions is considered. A necessary and sufficient condition for the existence of a unique solution for such a boundary value problem is established. The condition established answers a fundamental question for well-posedness and can be easily applied using a simple and computationally tractable algorithm that does not require finding the roots of the associated characteristic equation.

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