arXiv research

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnac…

In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear $f$-…

In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form $u_t = [E(x, u)u_x]_x + H(x, u)$ are studied. The point symmetry analysis of these equations is considered as well as an equivalence classification which admits an extension by one dimension of the principal Lie alge…

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such estimate on a Kähler manifold with a fixed Kähler metric. Then we consider the estimate on Kähler manifolds with Kähler metrics evolving under the rescaled Kähler-Ricci flow. Both of the estima…

The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.

New Brownian motion defined in Minkowski normed spaces.

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $u_t-Δu=au\log u+Vu, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth function on $M$ with $-…

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $ω_t=Δω+aω\ln ω$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $ω_t=Δω-ω\lnω+\varepsilon Rω$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove…

A preliminary group classification of the class 2D nonlinear heat equations $u_t=f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})$, where $f$ is arbitrary smooth function of the variables $x,y,u,u_x$ and $u_y$ using Lie method, is given. The paper is one of the few applications of an algebraic approach to the problem of group classific…

We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on …

Study proves energy critical heat equation solutions are Type I blowups for n ≥ 7.

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac…

There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE systems, and its concrete investigation is developing for the heat equation case. Th…

The purpose of this paper is to study gradient estimate of Hamilton - Souplet - Zhang type for the general heat equation $u_t=Δ_V u + au\log u+bu$ on noncompact Riemannian manifolds. As its application, we show a Harnak inequality for the heat solution and a Liouville type theorem for a nonlinear elliptic equation.…

Novel numerical scheme for G-heat equation with uncertainty.

This note introduces a regression technique for finding a class of nonlinear integro-differential operators from data. The method parametrizes the spatial operator with neural networks and Fourier transforms such that it can fit a class of nonlinear operators without needing a library of a priori selected operators. We…

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $Δu^{m}=δ(x)u_{t}+ψu^{m}$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on …

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

The paper characterizes stochastic incompleteness in Riemannian manifolds.

Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $u_t=Δ_f u+au\log u+bu +Au^p+Bu^{-q}$ and \[ u_t=Δ_f u+Ae^{pu}+Be^{-pu}+D \] are studied. As by-product, we obt…

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-ty…

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat …

In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive solutions on closed manifolds under Ricci flow is abtained. As applications, gra…

In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove …

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…

The paper establishes new inequalities for Finsler measure spaces.

Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type $$\ppt f=Δf-f \ln f +Rf.$$ We also comment on an earlier result of the first author on positive solutions of the c…

We propose a new cognitive framework for option price modelling, using quantum neural computation formalism. Briefly, when we apply a classical nonlinear neural-network learning to a linear quantum Schrödinger equation, as a result we get a nonlinear Schrödinger equation (NLS), performing as a quantum stochastic filter…

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…

Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space $H_1$. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Mar…

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…

Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equatio…

The paper studies harmonic map heat flow stability and decay rates.

We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stab…

A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock…

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=Δφ(u)$, $φ$ being an ar…

New heat equation method solves intertwining problems in CR geometry.

Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.

Bounds on Hessian of heat equation coupled with Ricci flow.

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $u_t=Δu^m$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curv…

This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.

In this paper, we study the Poisson equation and heat equation in a model matrix geometry $M_n$. Our main results are about the Poisson equation and global behavior of the heat equation on $M_n$. We can show that if $c_0$ is the initial positive definite matrix in $M_n$, then $c(t)$ exists for all time and is positive …

Estimates heat equation on shrinking Ricci solitons with uniform bounds.

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

Article proves Liouville theorem for heat equation in super Ricci flow.

Extends gradient estimates for heat equation under Finsler geometric flows.