Related papers: Liouville type theorems for dual nonlocal evolutio…
In this paper, we first study the dual fractional parabolic equation \begin{equation*} \partial^\alpha_t u(x,t)+(-\Delta)^s u(x,t) = f(u(x,t))\ \ \mbox{in}\ \ B_1(0)\times\R , \end{equation*} subject to the vanishing exterior condition. We…
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum…
In this paper, we study the fully fractional master equation \begin{equation}\label{pdeq1} (\partial_t-\Delta)^s u(x,t) =f(x,t,u(x,t)),\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}. \end{equation} First we prove a Liouville type theorem for…
We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive…
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and…
In this paper, we prove some Liouville theorem for the following elliptic equations involving nonlocal nonlinearity and nonlocal boundary value condition $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u(y)=\intpr \frac{…
This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-\Delta u=u^p+M|\nabla u|^q$ in $\Omega\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the…
A fundamental theorem of Liouville asserts that positive entire harmonic functions in Euclidean spaces must be constant. A remarkable Liouville-type theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of $-\Delta u=u^{…
We investigate Liouville-type results, existence, uniqueness and symmetry to the solution of nonlinear nonlocal elliptic equations of the form \[ Lu = |x|^{\gamma}\,H(u)\,G(\nabla u), \qquad x\in\R^n, \] where $L$ is a symmetric,…
This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation \begin{equation*} (\partial_t-\Delta)^s u(x,t) =f(u(x, t)), \,\,(x, t)\in B_1(0)\times \mathbb{R}, \end{equation*} subject to the…
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and…
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…
In this paper, we are concerned with the fractional and higher order H\'{e}non-Hardy type equations \begin{equation*} (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n}, \,\,\,…
We consider the fractional elliptic inequality with variable-exponent nonlinearity $$ (-\Delta)^{\frac{\alpha}{2}} u+\lambda\, \Delta u \geq |u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$, $\alpha\in (0,2)$, $\lambda\in\mathbb{R}$…
We establish a Liouville type theorem for fully nonlinear uniformly elliptic equations in exterior domains in half spaces under quadratic boundary data and a quadratic growth condition, that is, any viscosity solution tends to a quadratic…
We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$…
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…
We establish Liouville type theorems in the whole space and in a half-space for parabolic problems without scale invariance. To this end, we employ two methods, respectively based on the corresponding elliptic Liouville type theorems and…
In this paper, we prove the following result. Let $\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$ \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\…
In this paper, we establish Liouville type theorems for stable solutions on the whole space $\mathbb R^N$ to the fractional elliptic equation $$(-\Delta)^su=f(u)$$ where the nonlinearity is nondecreasing and convex. We also obtain a…