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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 principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $mathbb{R}_+^n times mathbb{R}$ and obtain some new connections between the nonexistence of solutions in a half space $mathbb{R}_+^n times mathbb{R}$ and in the whole space $mathbb{R}^{n-1} times mathbb{R}$ and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $lambdabeta e^{beta u }$, forced by an additive space-time white noise. We prove local and g
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obta
This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-triv
We study fractional parabolic equations with indefinite nonlinearities $$ frac{partial u} {partial t}(x,t) +(-Delta)^s u(x,t)= x_1 u^p(x, t),,, (x, t) in mathbb{R}^n times mathbb{R}, $$ where $0<s<1$ and $1<p<infty$. We first prove that all positive
We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.