We consider the Liouville system \begin{align*} -\Delta u = \lambda e^v , \quad -\Delta v = \mu e^u \quad \text{in } B \end{align*} with $u=v=0$ on $\partial B$, where $B$ is the unit ball in $\mathbb R^N$, $N\geq 3$, and $\lambda$ and $\mu$ are positive parameters. First we show that radial solutions in $B \setminus \{0\}$ are either regular or have a $\log$-type singularity. Then, in dimensions $3 \leq N \leq 9$ we prove that there is an unbounded curve $\mathcal S \subset (0,\infty)^2$ such that for each $(\mu,\lambda) \in\mathcal S$ there exist infinitely many regular solutions. Moreover, the number of regular solutions tends to infinity as $(\mu,\lambda)$ approaches a fixed point in $\mathcal S$.

We study the existence of solutions of a class of singularly perturbed BVPs $\varepsilon y'' +2y' +f(y) =0, ~y(0)=0, ~y(A)=0$ for some $A>0$ and $f>0$. Given an $f$ satisfying certain conditions, we will show that for each $\varepsilon>0$, there exists $A(\varepsilon)$ such that, if $0 < A < A(\varepsilon)$, then the problem has exactly two solutions. As an example we consider $f(y)=e^y$, where we show that $A(\varepsilon)>1$ for $\varepsilon>0$ sufficiently small. Both solutions have the same outer solution on $(0,1]$, but have different boundary behavior near $x=0$. One of them is bounded, the other unbounded as $\varepsilon \to 0$. A uniform expansion for the smaller solution has been obtained from matched asymptotic expansions. We prove rigorously that these asymptotics are correct up to $O(\varepsilon).$ We also prove that for a class of functions, $\underset{\varepsilon \to 0}{\lim} A(\varepsilon) = 2\int_0^{\infty} dy/f(y)$.

We study existence and regularity properties of solutions to the singular $p$-Laplacian parabolic system in a bounded domain $\Omega$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(\Omega))$, $\varepsilon\geq0$, integrability properties of the second spatial derivatives and of the time derivative of the solutions. Hence, for suitable $p$ and exponents $r$ and $q$, by Sobolev embedding theorems, we deduce global regularity of $u$ and $\nabla u$ in Hölder spaces. Finally we prove a global pointwise bound for the solution under the assumption $p>\frac{2n}{n+2}$.

In this paper we study the nonconvex anisotropic mean curvature flow of a hypersurface. This corresponds to an anisotropic mean curvature flow where the anisotropy has a nonconvex Frank diagram. The geometric evolution law is therefore forward-backward parabolic in character, hence ill-posed in general. We study a particular regularization of this geometric evolution, obtained with a nonlinear version of the so-called bidomain model. This is described by a degenerate system of two uniformly parabolic equations of reaction-diffusion type, scaled with a positive parameter $\epsilon$. We analyze some properties of the formal limit of solutions of this system as $\epsilon \to 0^+$, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented.

This work is devoted to proving the local null controllability of the Burgers-$\alpha$ model. The state is the solution to a regularized Burgers equation, where the transport term is of the form $zy_x$, $z=(Id-\alpha^2\frac{\partial^2}{\partial x^2})^{-1}y$, and $\alpha>0$ is a small parameter. We also prove some results concerning the behavior of the null controls and associated states as $\alpha\to 0^+$.

We consider the class of operators \[ Lu=\sum_{i,j=1}^{q}a_{ij}(x)X_{i}X_{j}u, \] where $X_{1},X_{2},\dots,X_{q}$ are homogeneous left-invariant Hörmander's vector fields on $\mathbb{R}^{N}$ with respect to a structure of Carnot group, $q\leq N,$ the matrix $\{ a_{ij}\} $ is symmetric and uniformly positive on $\mathbb{R}^{q},$ the coefficients $a_{ij} $ belong to $L^{\infty}\cap VLMO_{loc}( \Omega) $ ("vanishing logarithmic mean oscillation") with respect to the distance induced by the vector fields (in particular, they can be discontinuous), and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$. We prove local estimates in $BMO_{loc}\cap L^{p}$ of the following kind: \begin{align*} & \Vert X_{i}X_{j}u\Vert _{BMO_{loc}^{p}( \Omega^{\prime }) }+\Vert X_{i}u\Vert _{BMO_{loc}^{p}( \Omega ^{\prime}) } \\ & \leq c\big\{ \Vert Lu\Vert _{BMO_{loc}^{p}( \Omega) }+\Vert u\Vert _{BMO_{loc}^{p}( \Omega) }\big\} \end{align*} for any $\Omega^{\prime}\Subset\Omega$, $1 < p < \infty$. Even in the uniformly elliptic case $X_{i}=\partial_{x_{i}}$, $q=N$ our estimates improve the known results.