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This paper deals with a 1 / κ α -type length-preserving nonlocal flow of convex closed plane curves for all α > 0 . Under this flow, the convexity of the evolving curve is preserved. For a global flow, it is shown that the evolving curve converges smoothly to a circle as t → ∞ . Some numerical blow-up examples and a sufficient condition leading to the global existence of the flow are also constructed.

We study the oscillation behavior of solutions to the heat equation on R n and give some interesting examples. We compare the oscillation behavior of the initial data and the oscillation behavior of the solution as t → ∞ .

We look for solutions \begin{document}$ u\left( x,t\right) $\end{document} of the one-dimensional heat equation \begin{document}$ u_{t} = u_{xx} $\end{document} which are space-time periodic, i.e. they satisfy the property \begin{document}$ u\left( x+a,t+b\right) = u\left( x,t\right) $\end{document} for all \begin{document}$ \left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), $\end{document} and derive their Fourier series expansions. Here \begin{document}$ a\geq0,\ b\geq 0 $\end{document} are two constants with \begin{document}$ a^{2}+b^{2}>0. $\end{document} For general equation of the form \begin{document}$ u_{t} = u_{xx}+Au_{x}+Bu, $\end{document} where \begin{document}$ A,\ B $\end{document} are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when \begin{document}$ B>0 $\end{document} and is given by a linear combination of \begin{document}$ \cos\left( \sqrt{B}\left( x+At\right) \right) $\end{document} and \begin{document}$ \sin\left( \sqrt{B}\left( x+At\right) \right). $\end{document}

We use two simple methods to derive four important explicit graphical solutions of the curve shortening flow in the plane. They are well-known as the circle, hairclip, paperclip, and grim reaper solutions of the curve shortening flow. By the methods, one can also see that the hairclip and the paperclip solutions both converge to the grim reaper solutions as t → −∞.

This paper deals with a $$1/\kappa $$-type nonlocal flow for an initial convex closed curve $$\gamma _{0}\subset {\mathbb {R}}^{2}$$ which preserves the convexity and the integral$$\ \int _{X\left( \cdot ,t\right) }\kappa ^{\alpha +1}ds,\ \alpha \in \left( -\infty ,\infty \right) ,$$ of the evolving curve $$X\left( \cdot ,t\right) $$. For$$\ \alpha \in [1,\infty ),\ $$it is proved that this flow exists for all time $$t\in [0,\infty )$$ and $$X(\cdot ,t)$$ converges to a round circle in $$C^{\infty }$$ norm as $$t\rightarrow \infty $$. For $$\alpha \in \left( -\infty ,1\right) $$, a discussion on the possible asymptotic behavior of the flow is also given.

We study the oscillation behavior of solutions to the one-dimensional heat equation and give some interesting examples. We also demonstrate a simple ODE method to find explicit solutions of the heat equation with certain particular initial conditions.

We discuss several examples of curvature flows for convex closed plane curves that preserve parallel curves and use this property to find singularities (curvature blow-up) of these flows. A precise curvature blow-up rate is also obtained.

This paper deals with the 1 / κ α -type area-preserving nonlocal flow of smooth convex closed plane curves for all constant α > 0 . Under this flow, the convexity of the evolving curve is preserved. Due to the existence of finite time curvature blow-up examples, it is shown that, if the curvature κ will not blow up in finite time, the evolving curve will converge smoothly to a circle as t → ∞ .

For any $$\alpha >0,$$ we study $$k^{\alpha }$$ -type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $$C^{\infty } $$ -norm. Other relevant $$k^{\alpha }$$ -type nonlocal flow is also discussed when $$\alpha \ge 1.$$

We discuss several examples of non-parabolic curve flows in the plane. In these flows, the speed functions do not involve the curvature at all. Although elementary in nature, there are some interesting properties. In particular, certain non-parabolic flows can be employed to evolve a convex closed curve to become circular or to evolve a non-convex curve to become convex eventually, like what we have seen in the classical curve shortening flow (parabolic flow) by Gage and Hamilton (J Differ Geom 23:69–96, 1986), Grayson (J Differ Geom 26:285–314, 1987).

We study the asymptotic behavior of a nonlinear parabolic equation arising from anisotropic plane curve evolution. We show that, if the solution has type-one blow-up, then it will converge (after rescaling) to a self-similar solution. If the solution has type-two blow-up, then its profile near the maximum point is also known. In the case of anisotropic plane curve evolution, if the evolving curve has type-two blow-up, then it will converge to a translational self-similar solution.

We consider the expansion of a convex closed plane curve C0 along its outward normal direction with speed G(1/k), where k is the curvature and \({G \left(z \right) :\left(0, \infty \right) \rightarrow \left( 0, \infty \right)}\) is a strictly increasing function. We show that if \({{\rm lim}_{z \rightarrow \infty} G \left(z \right) = \infty}\), then the isoperimetric deficit \({D \left(t \right) : = L^{2}\left(t \right) -4 \pi A \left(t \right)}\) of the flow converges to zero. On the other hand, if \({{\rm lim}_{z \rightarrow \infty}G \left(z \right) = \lambda \in (0,\infty)}\), then for any number d ≥ 0 and \({\varepsilon > 0}\), one can choose an initial curve C0 so that its isoperimetric deficit \({D \left(t \right)}\) satisfies \({\left \vert D \left(t \right) -d \right \vert < \varepsilon}\) for all \({t \in [0, \infty)}\). Hence, without rescaling, the expanding curve Ct will not become circular. It is close to some expanding curve Pt, where each Pt is parallel to P0. The asymptotic speed of Pt is given by the constant \({\lambda}\).

We study a curve flow for convex closed plane curves. It is described by a third order linear equation for the radius of curvature of the evolving curve. It is shown that under the flow the evolving curve stays convex, bounds fixed area, length, and has fixed center. However, its curvature may blow up in finite time. ¶ If the curvature of this flow does not blow up before time $2\pi$, then the flow will exist smoothly for all time and is periodic in time with period $2\pi$. In particular, the flow does not have a limiting curve unless the initial curve is a circle.

We shall apply certain inequalities by Andrews and Green-Osher to area-preserving and length-preserving flows of convex plane curves and show that, if we have bound on the curvature, the evolving curves will converge to a round circle.

We study the contraction of a convex immersed plane curve with speed (1/{\alpha})k^{{\alpha}}, where {\alpha}in(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when {\alpha}=1), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson [GR])., Comment: 24 pages, 3 figures

We use the Aleksandrov reflection result of Chow and Gulliver to show that the center of expansion in expanding a given convex embedded closed curve γ 0 C ℝ 2 lies on a certain convex plane region interior to γ 0 .

For a class of general quasilinear equations on S 1 , we show that, by a very simple maximum principle technique, as long as the solution stays finite, all of its derivatives also remain finite. Some specific examples are given. Under suitable assumptions, we also derive exponential decay of the derivatives of the solution.

We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also have estimate on the curvature blow-up rate.

We show that a nontrivial homothetic self-similar solution can happen only when F ( k ) = k α or F ( k ) = − k − α . We also derive a parametric representation of a translational self-similar solution. A translational self-similar solution may have self-intersections but cannot be a simple closed curve for any F ( k ) .

We show that for embedded or convex plane curves expansion, the difference u(x,t)-r(t) in support functions between the expanding curves γ t and some expanding circles C t (with radius r(t)) has its asymptotic shape as t→∞. Moreover the isoperimetric difference L2-4πA is decreasing and it converges to a constant $\mathfrak{S} > 0$ if the expansion speed is asymptotically a constant and the initial curve is not a circle. For convex initial curves, if the expansion speed is asymptotically infinite, then L2-4πA decreases to $\mathfrak{S}=0$ and there exists an asymptotic center of expansion for γ t .

Motivated by a recent curvature flow introduced by Professor S.-T. Yau [S.-T. Yau, Private communication on his “Curvature Difference Flow”, 2007], we use a simple curvature flow to evolve a convex closed curve to another one (under the assumption that both curves have the same length). We show that, under the evolution, the length is preserved and if the curvature is bounded above during the evolution, then an initial convex closed curve can be evolved to another given one.