We investigate some well-posedness issues for the initial value problem associated to the system \begin{equation*} \begin{cases} u_{t}+\partial_x^3u+\partial_x(u^2v^3) =0,\\ v_{t}+\partial_x^3v+\partial_x(u^3v^2)=0, \end{cases} \end{equation*} for given data in low order Sobolev spaces $H^s(\mathbb R)\times H^s(\mathbb R)$. We prove local and global well-posedness results utilizing the sharp smoothing estimates associated to the linear problem combined with the contraction mapping principle. For data with small Sobolev norm we obtain global solution whenever $s\geq 0$ by using global smoothing estimates. In particular, for data satisfying $\delta < \|(u_0, v_0)\|_{L^2\times L^2} < \|(S, S)\|_{L^2\times L^2}$, where $S$ is solitary wave solution, we get global solution whenever $s>3/4$. To prove this last result, we apply the splitting argument introduced by Bourgain [5] and further simplified by Fonseca, Linares and Ponce [6, 7].

This paper investigates the boundary behavior of nonnegative classical solutions to the Dirichlet problem for $$ u_t=u^p\Delta u + g(u) \qquad \mbox{in } \Omega \times (0,T), \qquad p>1, $$ and draws some consequences for the large time behavior of solutions. Here, $\Omega \subset \mathbb R^n$ is a smooth bounded domain and $g:\ [0,\infty) \to \mathbb R$ is locally Lipschitz continuous with $g(0)=0$. The first goal is to study for which $\alpha>0$ the implication \begin{align} & & u_0(x) \le c_1 ({{\rm dist} \, } (x,{\partial\Omega}))^\alpha \qquad (c_1>0) \notag \\ & \Rightarrow & u(x,t) \le C(T') ({{\rm dist} \, }(x,{\partial\Omega}))^\alpha \quad \mbox{in } \Omega\times (0,T') \ \mbox{for any } T' < T, \qquad \mbox{(I)} \tag*{(I)} \end{align} is valid, and it turns out that this holds whenever either $p \ge 2$, or $p < 2$ and $\alpha \ge \frac{1}{p-1}$. For $p \in (1,2]$ and $g\equiv 0$, this complements a previously known result, according to which the lower estimate $u_0(x)\ge c_0(\text{dist}\, (x,{\partial\Omega}))^\alpha$ with some $\alpha <\frac{1}{p-1}$ and $c_0>0$ implies the existence of $T>0$ and $C>0$ such that $u(x,t) \ge C \text{dist}\, (x,{\partial\Omega})$ for all $x \in \Omega$ and $t \ge T$. Using (I), we moreover show that whenever $p>1$, there exist some values of $q \ge 1$ such that the particular equation, $u_t=u^p u_{xx}+u^q$, possesses positive classical solutions which are nondecreasing w.r.~to $t$ and remain uniformly bounded in $C^1(\bar\Omega)$ for all times, but do not converge in $C^1(\bar\Omega)$ as $t\to\infty$.

In this paper we prove the global existence and uniqueness of the low regularity solutions to the Cauchy problem of quasi-linear wave equations with radial symmetric initial data in three space dimensions. The results are based on the end-point Strichartz estimate together with the characteristic method.

In this paper, we analyze the symmetry properties of maximizers of a Hénon type functional in dimension two. Namely, we study the symmetry of the functions that realize the maximum $$ \sup_{u\in {{H^{1}(\Omega)}}\atop ||u||\le 1} {{\int_\Omega}} \left(e^{\gamma u^2} - 1\right)|x|^\alpha {{\,dx}}, $$ where $\Omega$ is the unit ball of $\mathbb R^2$ and $\alpha,\, \gamma>0$. We identify and study the limit functional $$ \sup_{u\in {{H^{1}(\Omega)}}\atop ||u||\le 1} {{\int_{\partial\Omega}}} \left(e^{\gamma u^2} - 1\right)\,ds, $$ which is the main ingredient to describe the behavior of maximizers as $\alpha\to \infty$. We also consider the limit functional as $\alpha\to 0$ and the properties of its maximizers.

We prove an intrinsic Harnack inequality for non-negative local weak solutions of a wide class of doubly nonlinear degenerate parabolic equations whose prototype is \begin{equation*} u_t-\mathrm{div}(u^{m-1}|Du|^{p-2}Du)=0,\qquad p{\geqslant} 2, m{\geqslant} 1. \end{equation*} As a consequence, we get that such solutions are locally Hölder continuous.

We show that the introduction of third and fourth order anisotropic dispersion terms in the nonlinear Schrödinger equation enables to solve the global Cauchy problem for nonlinearities which are (super) critical in the standard case. Some blow-up results are proved in the supercritical case. Then, we prove the existence of solitary waves and discuss their regularity, their exponential decay rate at infinity, and their stability.

Due to an oversight in the final production process, Figures 2 and 4 in [1] only showed part of the intended contents. We provide below the complete version for Figures 2 and 4.