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In this article, we are concerned with the tempered fractional parabolic problem 
$$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$
where 
$-\left (\Delta +\lambda \right )^{\frac {\alpha }{2}}$ is a tempered fractional operator with 
$\alpha \in (0,2)$ and 
$\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators.
In this work, we obtain a local maximum principle along the Ricci flow 
$g(t)$
under the condition that
$\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$
for
$t>0$
for some constant
${\alpha }>0$
. As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for
$t>0$
, provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for
$t>0$
.
We study the non-autonomously forced Burgers equation
on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.
$$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$
In this paper, we analyse the semilinear fourth-order problem ( − Δ)2u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and
for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu > 0 in ℝN, and they do not exist when the previous condition fails.
$$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$
We consider an elliptic problem of the type
where Ω is a bounded Lipschitz domain in ℝN with a cylindrical symmetry, ν stands for the outer normal and 
$$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$
$\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $.
Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.
As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem
For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L2 (Ω) × L2(Γ2).
$$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$
