Related papers: Q-curvature flow with indefinite nonlinearity
The main result of this paper is: Given any constant C, there is $(\epsilon,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(\epsilon,k,L)$-neck, then the…
We produce longtime solutions to the K\"ahler-Ricci flow for complete K\"ahler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded…
We consider a general curvature equation $F(\kappa)=G(X,\nu(X))$, where $\kappa$ is the principal curvature of the hypersurface $M$ with position vector $X$. It includes the classical prescribed curvature measures problem and area measures…
We produce solutions to the K\"ahler-Ricci flow emerging from complete initial metrics $g_0$ which are $C^0$ Hermitian limits of K\"ahler metrics. Of particular interest is when $g_0$ is K\"ahler with unbounded curvature. We provide such…
We consider the Paneitz-type equation $\Delta^2 u -\alpha \Delta u +\beta (u-u^q ) =0$ on a closed Riemannian manifold $(M,g)$. We reduce the equation to a fourth-order ordinary differential equation assuming that $(M,g)$ admits a proper…
A little complement concerning the dynamics of non-metric manifolds is provided, by showing that any flow on an $\omega$-bounded surface with non-zero Euler character has a fixed point.
For an infinite Toeplitz matrix $T$ with nonnegative real entries we find the conditions, under which the equation $\boldsymbol{x}=T\boldsymbol{x}$, where $\boldsymbol{x}$ is an infinite vector-column, has a nontrivial bounded positive…
We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-\Delta)^\frac{n}{2}w=e^{nw}-c\delta_{0} \text{ on } \mathbb R^n,$$ under a finite volume…
The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being…
We study the $O(N)$ non-linear $\sigma$ model on three-dimensional manifolds of constant curvature by means of the large $N$ expansion at the critical point. We examine saddle point equations imposing anti-periodic boundary condition in…
Consider an arbitrary closed, countably $n$-rectifiable set in a strictly convex $(n+1)$-dimensional domain, and suppose that the set has finite $n$-dimensional Hausdorff measure and the complement is not connected. Starting from this given…
We consider a finite difference approximation of mean curvature flow for axisymmetric surfaces of genus zero. A careful treatment of the degeneracy at the axis of rotation for the one dimensional partial differential equation for a…
We prove the existence and uniqueness, up to a shift in time, of curved traveling fronts for a reaction-advection-diffusion equation with a combustion-type nonlinearity. The advection is through a shear flow $q$. This analyzes, for…
Linear stability of inviscid, parallel, and stably stratified shear flow is studied under the assumption of smooth strictly monotonic profiles of shear flow and density, so that the local Richardson number is positive everywhere. The…
We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*}…
Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various…
In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $\lambda$ of the maximum and the minimum of the holomorphic sectional curvatures…
We investigate a model equation in the crystal growth, which is described by a level-set mean curvature flow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed.…
We revisit the well-known Curve Shortening Flow for immersed curves in the $d$-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is…
In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii's fixed point theorem on cones, sufficient conditions for the…