Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

2018 ◽  
Vol 2020 (23) ◽  
pp. 9440-9470
Author(s):  
Jian Lu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Smooth Function ◽  
Existence Result ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

Abstract In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}$$is studied, where $f$ is a positive $2\pi $-periodic smooth function. By virtue of a new generalized Blaschke–Santaló inequality, we obtain a new existence result of solutions.

A note on the Abresch—Langer conjecture

2014 ◽  
Vol 144 (2) ◽  
pp. 299-304 ◽  
Author(s):  
Kai-Seng Chou ◽  
Xiao-Liu Wang
Keyword(s):  
Saddle Point ◽  
The Self ◽  
Point Property ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions

A saddle-point property of the self-similar solutions in the curve shortening flow was conjectured by Abresch and Langer and confirmed by Au. An improvement on Au's solution is presented.

scholarly journals Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Yongxiang Li ◽  
Lanjun Guo
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Existence Result ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Superlinear Growth ◽  
Existence Of Periodic Solutions

This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation u′′(t)=ft,ut,u′t, t∈R, where the nonlinearity f:R3→R is continuous and f(t,x,y) is 2π-periodic in t. Under certain inequality conditions that f(t,x,y) may be superlinear growth on (x,y), an existence result of odd 2π-periodic solutions is obtained via Leray-Schauder fixed point theorem.

scholarly journals On self-similar solutions of time and space fractional sub-diffusion equations

2021 ◽  
Vol 11 (3) ◽  
pp. 16-27
Author(s):  
Fatma Al-Musalhi ◽  
Erkinjon Karimov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Derivatives ◽  
Essential Role ◽  
Differential Operators ◽  
Diffusion Equations ◽  
Fractional Partial Differential Equation ◽  
Successive Iteration ◽  
Self Similar ◽  
Similar Solutions

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.

scholarly journals TRANSLATING SOLITONS FOR LAGRANGIAN MEAN CURVATURE FLOW IN COMPLEX EUCLIDEAN PLANE

2012 ◽  
Vol 23 (10) ◽  
pp. 1250101 ◽  
Author(s):  
ILDEFONSO CASTRO ◽  
ANA M. LERMA
Keyword(s):  
Mean Curvature ◽  
Euclidean Plane ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Curve Shortening Flow ◽  
Self Similar ◽  
Curve Shortening ◽  
Similar Solutions ◽  
Hamiltonian Stationary ◽  
Translating Solitons

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones [Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom.84 (2010) 127–161] in dimension two. The simplest (non-trivial) example in our family is characterized as the only (non-totally geodesic) Hamiltonian stationary Lagrangian translating soliton for mean curvature flow in complex Euclidean plane.

The G-invariant implicit function theorem in infinite dimensions

1982 ◽  
Vol 92 (1-2) ◽  
pp. 13-30 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Implicit Function Theorem ◽  
Implicit Function ◽  
Infinite Dimensions ◽  
Basic Theorem

SynopsisOur basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries. The result is sufficiently general to cover a great many applications. It generalizes some earlier work of the author and corrects and improves some work of Vanderbauwhede. We also consider the breaking of symmetries problem and the variational case. Finally, we apply our results to study the periodic solutions of an ordinary differential equation.

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