We study the equation $- \Delta u + |x|^a |u|^{q-2} u = |x|^b |u|^{p-2} u$ with Dirichlet boundary condition on $B(0,1)$ or on $\mathbb R^N$. We study the radial solutions of this equation on~$\mathbb R^N$ and the symmetry breaking for ground states for $q=2$ on $\mathbb R^N$. Estimates of the transition are also given when $p$ is close to $2$ or $2^*$ on $B(0,1)$.

A class of nonlinear integro-differential Cauchy problems is studied by means of the viscosity solutions approach. In view of financial applications, we are interested in continuous initial data with exponential growth at infinity. Existence and uniqueness of solution is obtained through Perron's method, via a comparison principle; besides, a first order regularity result is given. This extension of the standard theory of viscosity solutions allows to price derivatives in jump--diffusion markets with correlated assets, even in the presence of a large investor, by means of the PDE's approach. In particular, derivatives may be perfectly hedged in a completed market.

In this paper we describe the asymptotic behavior of the positive solutions of a class of parabolic equations according to the size of a certain parameter. Within the range of values of the parameter where the model does not admit an attracting classical steady state it possesses an attracting metasolution ---a very weak generalized solution. It turns out that the minimal metasolution attracts all positive solutions starting in a subsolution and that the limiting profile of any other positive solution lies in the order interval defined by the minimal and the maximal metasolution.

A linear elliptic equation with Neumann boundary conditions and unbounded drift coefficients is studied. The existence and uniqueness results are used to characterize the infinitesimal generator of the transition semigroup associated with a stochastic variational inequality.

We consider the time-local well-posedness for the initial-value problem of the fourth-order nonlinear Schrödinger-type equation in one space dimension which describes the motion of the vortex filament. By using the method of Fourier restriction norm introduced by Bourgain [3] and Kenig-Ponce-Vega [17]--[19], we show the time-local well-posedness in the Sobolev space $H^s(\mathbb R)$ with $s\ge1/2$ under certain coefficient conditions.

We consider a coupled system of partial differential equations (PDEs) of hyperbolic/parabolic type, which is a generalization of an established structural acoustic model. A peculiar feature of this model is a high damping term in the boundary condition of the wave component which accounts for lack of uniform stability of the overall system, even in the presence of viscous damping on the entire domain. It has been shown recently that by introducing a comparable static damping in the boundary condition of the wave component, the corresponding feedback system is uniformly stable. In this paper we study the stability properties of the coupled system when the internal damping is subject to nonlinear effects. We provide two main different stability results, which describe decay rates of the solutions to the coupled PDE system via the solution to an appropriate nonlinear ordinary differential equation. Our analysis allows saturation of the nonlinear dissipation term. In this significant case we obtain as well uniform decay estimates of the underlying energy, provided that initial data are measured with a slightly stronger topology. Even though the prime concern of this paper is to deal with coupled structures and the overdamping phenomenon, these results yield, as well, new results on uniform stabilization of a single wave equation.