Related papers: Liouville theorem for subcritical nonlinear heat e…

Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a…

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ \partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty…

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

We study a time-fractional semilinear heat equation $$\partial^{\alpha}_t u -\Delta u = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^{\alpha}$ denotes the…

In this paper we prove a Liouville type theorem for the stationary equations of a non-Newtonian fluid in $\mathbb{R}^3$ with the viscous part of the stress tensor $\mathbf{A}_p(u) = \mathrm{div} ( | \mathbf{D}(u) |^{p-2} \mathbf{D}(u) )$,…

We study the fourth order Schr\"odinger type differential inequality $-\Delta^2 u + \lambda V(x)u \geq a(x)u^q$ with $a,V\in L^1_{loc}(\mathbf{R}^N)$, both nonnegative, and $\lambda>0$. We consider nonnegative solutions without making any…

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex…

In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…

Let $p:\mathbb{C}\to\mathbb{R}$ be a subharmonic, nonharmonic polynomial and $\tau$ a real parameter. Define $\bar{Z}_{\tau p} = \partial_{\bar z} + \tau p_{\bar z}$, a closed, densely-defined operator on $L^2(\mathbb{C})$. If $\Box_{\tau…

In this paper, we combine Bochner formula, Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $\Delta_pu+\Delta_qu+h(u,|\nabla…

In this article we derive gradient estimation for positive solution of the equation \begin{equation*} (\partial_t-\Delta_f)u = A(u)p(x,t) + B(u)q(x,t) + \mathcal{G}(u) \end{equation*} on a weighted Riemannian manifold evolving along the…

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-\'Emery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation…

In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{E}mery curvature is…

In this paper, we derive the gradient estimates for the positive solutions of the equation $\Delta_b u + au^{p+1} = 0$ on complete noncompact pseudo-Hermitian manifolds, where $a > 0$ and $p \leq 0$ or $a < 0$ and $p > 0$ are two constants.…

We prove a theorem for the growth of the energy of bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form $\Delta u=\nabla W(u)$, $x\in \mathbb{R}^n$, $n\geq 2$, with $W:\mathbb{R}^m\to \mathbb{R}$,…

Gradient inequalities of the Hamilton type and the Li-Yau type for positive solutions to the heat equation are established from a probabilistic viewpoint, which simplifies the proofs of some results of Sun [{\it Pacific J. Math.}, 253…

In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic…

We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our…

We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type…

We use the integral by parts to get a Liouville type theorem for a class quasilinear $p$-Laplace type equation on the sphere, this $p$-Laplace type equation arises from the study of asymptotic behavior near the origin for the semi-linear…