Related papers: Counterexamples to quasiconcavity for the heat equ…
In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete non-compact Riemannian manifold M or Kahler-Ricci flow. We show that under a natural assumption, a new partial convexity…
In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly…
A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.
We construct examples for the one-phase Stefan problem which show that $\alpha$-concavity of the solution is in general not preserved in time, for $0 \le \alpha <1/2$. In particular, this shows that, in contrast to the case of the heat…
In this paper, an inverse initial-boundary value problem for the heat equation in three dimensions is studied. Assume that a three-dimensional heat conductive body contains several cavities of strictly convex. In the outside boundary of…
In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in…
We study the convexity of mutual information along the evolution of the heat equation. We prove that if the initial distribution is log-concave, then mutual information is always a convex function of time. We also prove that if the initial…
In this short paper we provide a new proof of the geometric Forward-Reverse Brascamp-Lieb inequality, using the approach of the heat semigroup, or the heat flow. Furthermore, we characterize all the Forward-Reverse Brascamp-Lieb data such…
We prove that the initial-value problem for the fractional heat equation admits a solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove…
An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary…
The heat equation does not have time-reversal invariance. However, using a solution of an associated wave equation which has time-reversal invariance, one can establish an explicit extraction formula of the minimum sphere that is centered…
The one-dimensional problem of the nonlinear heat equation is considered. We assume that the heat flow in the origin of coordinates is the power function of time and the initial temperature is zero. Approximate solutions of the problem are…
The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for…
We study the Cauchy problem for a semilinear heat equation with initial data non-rarefied at $\infty$. Our interest lies in the discussion of the effect of the non-rarefied factors on the life span of solutions, and some sharp estimates on…
In this article, we study the the harmonic map heat flow from a manifold with conic singularities to a closed manifold. In particular, we have proved the short time existence and uniqueness of solutions as well as the existence of global…
We show that, on a complete, connected and non-compact Riemannian manifold of non-negative Ricci curvature, the solution to the heat equation with $L^{1}$ initial data behaves asymptotically as the mass times the heat kernel. In contrast to…
We study the existence and uniqueness of source-type solutions to the Cauchy problem for the heat equation with fast convection under certain tail control assumptions. We allow the solutions to change sign, but we will in fact show that…
This paper deals with some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings. Constant boundary conditions are imposed on the single component of the boundary when the domain…
This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if…
Differentially rotating, "advection-dominated" accretion flows are considered in which the heat generated by viscous dissipation is retained in the fluid. The equations of time-dependent quasi-spherical accretion are solved in a simplified…