In this paper, we consider non-autonomous Ornstein-Uhlenbeck operators in smooth exterior domains $\Omega\subset \mathbb R^d$ subject to Dirichlet boundary conditions. Under suitable assumptions on the coefficients, the solution of the corresponding non-autonomous parabolic Cauchy problem is governed by an evolution system $\{P_\Omega(t,s)\}_{0\le s\le t}$ on $L^p(\Omega)$ for $1< p < \infty$. Furthermore, $L^p$-estimates for spatial derivatives and $L^p$-$L^q$ smoothing properties of $P_\Omega(t,s),\,0\le s\le t,$ are obtained.

We consider the heat equation in a polygonal domain $\Omega$ of the plane in weighted $L^p$-Sobolev spaces \begin{equation} \begin{array}{cl} \partial_t u -\Delta u = h, & \mbox{in } \Omega \times {(-\pi,\pi ) }, \\ u=0,& \mbox{on } \partial\Omega \times {[-\pi,\pi]}, \\ u(\cdot, -\pi)=u(\cdot, \pi),& \mbox{in } \Omega. \end{array} \tag*{(0.1)} \end{equation} Here $h$ belongs to $L^p(-\pi,\pi;L^p_\mu(\Omega))$, where $L^p_\mu(\Omega)=\{v \in L^p_{loc}(\Omega): r^\mu v\in L^p(\Omega)\},$ with a real parameter $\mu$ and $r(x)$ the distance from $x$ to the set of corners of $\Omega$. We give sufficient conditions on $\mu$, $p$, and $\Omega$ that guarantee that problem (0.1) has a unique solution $u\in L^p(-\pi,\pi;L^p_\mu(\Omega))$ that admits a decomposition into a regular part in weighted $L^p$-Sobolev spaces and an explicit singular part. The classical Fourier transform techniques do not allow one to handle such a general case. Hence we use the theory of sums of operators.

We consider the Cauchy problem for the Kawahara equation, which is a fifth-order KdV equation. This paper establishes the local well-posedness with initial data given in the Sobolev space $H^s(\mathbb{R})$. Previously, Chen, Li, Miao, and Wu (2009) proved the local well-posedness for $s>-7/4$, which has been improved to $s \geq -7/4$ by Chen and Guo. We improve this result to $s \geq -2$. The main idea is to modify the Bourgain space. Similar arguments are used by Bejenaru and Tao (2006). Moreover, we prove ill-posedness for $s<-2$ by using the argument by Bejenaru and Tao (2006).

We are concerned with a quasilinear elliptic equation of the form \begin{equation*} -\Delta u+a(x) u-\Delta(|u|^\alpha)|u|^{\alpha-2}u=h(u)\quad \hbox{in }\mathbb R^N, \end{equation*} where $\alpha> 1$ and $N\geq 1$. By using variational approaches, we prove the existence of at least one positive solution of the above equation under suitable conditions on $a(x)$ and $h$. In particular, we are interested in the situation that $a(x)$ is invariant under the finite group action $G$.

Let $n>2$ be even; $r\geq1$ be an integer; $0<\alpha<1$; $\Omega$ be a bounded, connected, smooth, open set in $\mathbb{R}^{n}$; and $\nu$ be its exterior unit normal. Let $f,g\in C^{r,\alpha}(\overline{\Omega};\Lambda^{2})$ be two symplectic forms (i.e., closed and of rank $n$) such that $f-g$ is orthogonal to the harmonic fields with vanishing tangential part, $\nu\wedge f,\nu\wedge g\in C^{r+1,\alpha}(\partial\Omega;\Lambda^{3})$ and $\nu\wedge f=\nu\wedge g$ on $\partial\Omega.$ Moreover assume that $tg+(1-t)f$ has rank $n$ for every $t\in\lbrack0,1].$ We will then prove the existence of a $\varphi \in\operatorname{Diff}^{r+1,\alpha}(\overline{\Omega};\overline{\Omega})$ satisfying \[ \left\{ \begin{array} [c]{cl} \varphi^{\ast}(g)=f & \text{in $\Omega$}\\ \varphi=\operatorname{id} & \text{on $\partial\Omega.$} \end{array} \right. \]

This paper is devoted to the study of the asymptotic behaviors of the minimal speed of propagation of pulsating travelling fronts solving the Fisher-KPP reaction-advection-diffusion equation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. We consider a periodic heterogenous framework and we use the formula of Berestycki, Hamel, and Nadirashvili [3] for the minimal speed of propagation to prove the asymptotics in any space dimension $N.$ We express the limits as the maxima of certain variational quantities over the family of ``first integrals'' of the advection field. Then, we perform a detailed study in the case $N=2$ which leads to a necessary and sufficient condition for the positivity of the asymptotic limit of the minimal speed within a large drift.