The purpose of this paper is to state some qualitative properties of the solutions to the Emden-Fowler equation $\Delta u + r^\sigma u^p = 0$ in a cone with Dirichlet boundary conditions. Namely one can show that every solution has the same symmetry as the cone in some sense; furthermore it is possible to extend the nonexistence results for regular solutions to this equation already stated by C. Bandle and M. Essen in [2]. For this one needs to establish some asymptotics for the solutions as $r\rightarrow 0$ or $r\rightarrow \infty$, relying on methods used by Veron in [25] for similar equations, but in different geometries, and then use a special form of the moving-planes method on a sphere in the spirit of [22].

Consider a nonlinear $m^{\text{th}}$ order evolution PDE \begin{equation} u_t = \mathbf A(u) \equiv A(x,u,u_1, \dots ,u_m), \quad u=u(x,t), \quad (x,t) \in Q=(0,1) \times [0,1], \tag*{(*)} \end{equation} where $A$ is a $C^\infty$ function and $u_t = \partial u/\partial t$, $u_i = \partial ^i u / \partial x^i$. If (*) is invariant under a group of scalings with the infinitesimal generator $ X =x \frac {\partial}{\partial x} + \mu t \frac {\partial}{\partial t} $ ($\mu$ is a constant ``scaling order" of $\mathbf A$), then the PDE admits exact self-similar solutions depending on the single invariant variable $u(x,t) = {\theta } (\xi)$, $ \xi = x/t^{1/\mu}$, where ${\theta }$ solves a nonlinear $m^{\text{th}}$ order ODE associated with the PDE. We prove that when the operator $\mathbf A$ is composed of a finite sum of operators with different scaling orders, $\mathbf A = \sum \mathbf A_i$, and no group of scalings exists, the exact solutions can be constructed via the invariance of the set $S_0 = \{u: u_1 =F(u)/x\}$ of a contact first-order differential structure, where $F$ is a smooth function to be determined. The time-evolution on $S_0$ is shown to be governed by a first-order dynamical system. We thus observe that besides scaling group properties, the invariance of $S_0$ specifies new sets of solutions described by first-order ODEs. The approach applies to a class of nonlinear parabolic equations of the second and of the fourth order.

In this paper we show the existence of non-topological 0-vortex and 1-vortex solutions for a nonlinear elliptic equation arising from Chern-Simons-Higgs theory in a general background metric $(g_{\mu\nu})=diag(1, -k(x), -k(x))$ with decay $k(x)=O(|x|^{-l})$ for some $ l >2$ at infinity.

In this paper we give solutions of boundary value problems for differential forms, generalizing Poisson's equation for functions. We focus on problems which for instance give an easy access to decomposition results like those of Hodge, Kodaira and Morrey. The search for general solutions and the proof of appropriate a priori estimates will be the central part of this publication.

In the present paper, we use some standard a priori estimates for linear transport equations to prove the existence and uniqueness of solutions for the Camassa-Holm equation with minimal regularity assumptions on the initial data. We also derive some explosion criteria and a sharp estimate from below for the existence time. We finally address the question of global existence for certain initial data. This yields, among other things, a different proof for the existence and uniqueness of Constantin and Molinet's global weak solutions (see [9]).

We consider the regularity of parabolic equations. We obtain that the solution belongs to Gevrey class 2 up to the boundary if functions in the equation belong to Gevrey class 2 in all dependent variables.

The Willmore flow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of solutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere, provided that they are initially close to a sphere.

We consider two-dimensional autonomous flows with divergence free continuous coefficients. Under a generic assumption of regularity on the set of critical points, we give a proof of uniqueness for the characteristics and for the transport equation in the framework of distributions.