Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace–Beltrami Equations on Spheres

2015 ◽  
Vol 58 (4) ◽  
pp. 723-729 ◽  
Author(s):  
Alfonso Castro ◽  
Emily M. Fischer
Keyword(s):  
Differential Equation ◽  
Phase Plane ◽  
Phase Plane Analysis ◽  
Point Boundary ◽  
Initial Value ◽  
Beltrami Equations ◽  
Plane Analysis ◽  
Singular Ordinary Differential Equation ◽  
Rotationally Symmetric ◽  
Pohozaev Type Identity

AbstractWe show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ℝn has inûnitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as |s|p-1s for |s| large with 1 < p < (n + 5)/(n − 3).

A NONOSCILLATION THEOREM FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DECAYING COEFFICIENTS

2001 ◽  
Vol 33 (3) ◽  
pp. 299-308 ◽  
Author(s):  
JITSURO SUGIE
Keyword(s):  
Differential Equation ◽  
Phase Plane ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Phase Plane Analysis ◽  
Nontrivial Solutions ◽  
Plane Analysis ◽  
Nonoscillation Theorem ◽  
Special Case ◽  
Fowler Equation

The purpose of this paper is to give sufficient conditions for all nontrivial solutions of the nonlinear differential equation x″ +a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfies the sign condition xg(x) > 0 if x ≠ 0, but is not assumed to be monotone increasing. This differential equation includes the generalized Emden–Fowler equation as a special case. Our main result extends some nonoscillation theorems for the generalized Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase-plane analysis.

A non-oscillation theorem for nonlinear differential equations with p-Laplacian

2006 ◽  
Vol 136 (3) ◽  
pp. 633-647 ◽  
Author(s):  
Jitsuro Sugie ◽  
Masakazu Onitsuka
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Linear Differential Equation ◽  
Phase Plane ◽  
Nonlinear Differential Equations ◽  
Phase Plane Analysis ◽  
Oscillation Theorem ◽  
Plane Analysis ◽  
Linear Differential

The equation considered in this paper is tp(φp(x′))′ + g(x) = 0, where φp(x′) = |x′|p−2x′ with p > 1, and g(x) satisfies the signum condition xg(x) > 0 if x ≠ 0 but is not assumed to be monotone. Our main objective is to establish a criterion on g(x) for all non-trivial solutions to be non-oscillatory. The criterion is the best possible. The method used here is the phase-plane analysis of a system equivalent to this differential equation. The asymptotic behaviour is also examined in detail for eventually positive solutions of a certain half-linear differential equation.

scholarly journals Qualitative study of anisotropic cosmological models with dark sector coupling

2017 ◽  
Vol 95 (11) ◽  
pp. 1049-1061 ◽  
Author(s):  
Rakesh Raushan ◽  
R. Chaubey
Keyword(s):  
Dark Energy ◽  
Phase Plane ◽  
Bianchi Type ◽  
Dynamical Evolution ◽  
Phase Plane Analysis ◽  
Type I ◽  
Dark Sector ◽  
Linear Coupling ◽  
Plane Analysis ◽  
Rotationally Symmetric

In this paper, we study the dynamical evolution of locally rotationally symmetric (LRS) Bianchi type I cosmological model with coupling of dark sector. We investigate the phase-plane analysis when dark energy is modelled as exponential quintessence, and is coupled to dark energy matter via linear coupling between both dark components. The evolution of cosmological solutions is studied by using dynamical systems techniques. Stability and viability issues for three different physically viable linear couplings between both dark components are presented and discussed in detail.

Growth of an Infinitesimal Cavity in a Rate-Dependent Solid

1989 ◽  
Vol 56 (1) ◽  
pp. 40-46 ◽  
Author(s):  
Rohan Abeyaratne ◽  
Hang-sheng Hou
Keyword(s):  
Phase Plane ◽  
Incompressible Material ◽  
Large Strains ◽  
Phase Plane Analysis ◽  
Initial Radius ◽  
Cavity Radius ◽  
Initial Value ◽  
Rate Dependent ◽  
Plane Analysis ◽  
Cavitation Region

This study examines the effect of rate dependence on growth of an infinitesimal cavity in a homogeneous, isotropic, incompressible material. Specifically, a sphere containing a traction-free void of infinitesimal initial radius is considered, its outer surface being subjected to a prescribed uniform radial nominal stress p, which is suddenly applied and then held constant. The sphere is composed of a particular class of rate-dependent materials. The large strains which occur in the vicinity of the void are accounted for in the analysis, and the problem is reduced to a nonlinear initial value problem, which is then studied qualitatively through a phase plane analysis. The principal results of this paper consist of two equations that are derived between the applied stress p and the cavity radius b: p = pˆ(b) and p = p(b). The first of these describes a curve which separates the (p, b)-plane into regions where cavitation does and does not occur. The second describes a curve which further subdivides the former subregion—the post-cavitation region—into domains where void expansion occurs slowly and rapidly.

Existence and nonexistence of limit cycles for Liénard-type equations with bounded nonlinearities and φ-Laplacian

2017 ◽  
Vol 19 (06) ◽  
pp. 1650057 ◽  
Author(s):  
Kōdai Fujimoto ◽  
Naoto Yamaoka
Keyword(s):  
Differential Equation ◽  
Limit Cycle ◽  
Nonlinear Differential Equation ◽  
Limit Cycles ◽  
Phase Plane ◽  
Sufficient Conditions ◽  
Phase Plane Analysis ◽  
Equivalent System ◽  
Plane Analysis ◽  
Existence And Nonexistence

This paper deals with an equivalent system to the nonlinear differential equation of Liénard type [Formula: see text], where the range of the function [Formula: see text] is bounded. Sufficient conditions are obtained for the system to have at least one limit cycle. The proofs of our results are based on phase plane analysis of the system with the Poincaré–Bendixon theorem. Moreover, to show that these sufficient conditions are suitable in some sense, we also establish the results that the system has no limit cycles. Finally, some examples are given to illustrate our results.

An Initial Value Method for the Solution of a Class of Nonlinear Equations in Fluid Mechanics

1970 ◽  
Vol 92 (3) ◽  
pp. 503-508 ◽  
Author(s):  
T. Y. Na
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Porous Medium ◽  
Fluid Mechanics ◽  
Boundary Layer Equation ◽  
Physical Parameters ◽  
Point Boundary ◽  
Trial And Error ◽  
Initial Value ◽  
Blasius Boundary Layer

An initial value method is introduced in this paper for the solution of a class of nonlinear two-point boundary value problems. The method can be applied to the class of equations where certain physical parameters appear either in the differential equation or in the boundary conditions or both. Application of this method to two problems in Fluid Mechanics, namely, Blasius’ boundary layer equation with suction (or blowing) and/or slip and the unsteady flow of a gas through a porous medium, are presented as illustrations of this method. The trial-and-error process usually required for the solution of such equations is eliminated.

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