Considered herein is the Korteweg-de Vries equation with a Kuramoto-Sivashinsky dissipative term appended. This evolution equation, which arises as a model for a number of interesting physical phenomena, has been extensively investigated in a recent paper of Ercolani, McLaughlin and Roitner. The numerical simulations of the initial-value problem reported in the just-mentioned study showed solutions to possess a more complex range of behavior than the unadorned Korteweg-de Vries equation. The present work contributes some basic analytical facts relevant to the initial-value problem and to some of the conclusions drawn by Ercolanet al. In addition to showing the initial-value problem is well posed, we determine the limiting behavior of solutions as the dissipative or the dispersive parameter tends to zero.

We consider the nonlinear equation $$ u_t = \Delta \log u $$ posed in two space dimensions. For the Cauchy problem with radially symmetric data, we investigate the existence of solutions, both global and local in time, as well as the question of uniqueness/multiplicity. The most striking result is as follows: for every radial $u(x,0)\in L^1(\mathbf{R}^2)$ there exists a unique maximal solution $u\in C^\infty(\mathbf{R}^2\times (0,T))$ of the Cauchy problem, characterized by the additional property \begin{equation} \int_{\mathbf{R}^2} u(x,t)\,dx= \int_{\mathbf{R}^2} u(x,0)\,dx -4\pi\,t\,, \tag{*} \end{equation} and, accordingly, the existence time is $T=\int u(x,0)\,dx/4\pi\,$. We then interpret the solutions as the conformal factor of a metric in $\mathbf{R}^2$ evolving by Ricci flow; formula (*) is a version of Gauss-Bonnet's Theorem. The solution here described is not unique if one weakens the equality (*) into an inequality $\le\,$. We thus obtain infinitely many nonmaximal solutions of the Cauchy problem having different behaviors (more precisely fluxes) at $r=+\infty\,$. One of these options, namely the solution corresponding to formula (*) with last term $-8\pi t$, describes the evolution of a complete compact surface under Ricci flow. For data $u(x,0)$ with infinite integral solutions are unique.

We establish comparison theorems for solutions of degenerate parabolic partial differential equations with discontinuities (or singularities). The results are used to define the generalized evolutions of sets generated by geometric partial differential equations with discontinuities. These results generalize recent results obtained by Ohnuma-Sato ([13]) and Gurtin-Soner-Souganidis ([11]).

We describe all nontrivial nonnegative solutions to the problem $$ \begin{cases} -\Delta u = a u^{{n+2}\over{n-2}} \quad &\hbox{in }\ H, \cr {{\partial u }\over {\partial \nu }} = bu^{{n}\over{n-2}} &\hbox{on } \ \partial H, \end{cases} $$ where $H$ is the half space of $\mathbb{R}^n (n\ge3)$.

The paper deals with the equation $\Delta u \pm |\nabla u|^q = f (u)$ in $\Omega \subset \mathbf{R}^n$, where $u$ blows up at the boundary $\partial \Omega$ and $\Omega$ is a bounded domain, which satisfies an interior and an exterior sphere condition. The existence and the asymptotic behaviour of $u$ near the boundary are investigated, showing how the nonlinear gradient term affects the results.