We analyze the perturbed Klein-Gordon Cauchy problem in two space dimension, studying the limit of mass decreasing to zero. We prove a global existence result when the initial data are small and the nonlinearity is at least quadratic near the origin.

We study the existence of solutions of stochastic differential equations with a state constraint depending on the time. We provide a necessary and sufficient characterization of closed, time depending constraints for which there exists a solution of a given stochastic differential equation. This characterization is given in terms of viscosity super- and subsolution of some suitable partial differentiable equations. The above property, called viability, is stated for both forward and backward stochastic differential equations.

This article is devoted to the study of the long-time behavior of solutions of the following 4th order parabolic system in a bounded smooth domain $\Omega\subset\subset{\mathbb R}^n$: $$ b\partial_t u=-{\Delta_x} (a{\Delta_x} u-\alpha\partial_t u-f(u)+\tilde g )\,, \tag*{(1)} $$ where $u=(u^1,\cdots,u^k)$ is an unknown vector-valued function, $a$ and $b$ are given constant matrices such that $a+a^*>0$, $b=b^*>0$, $\alpha>0$ is a positive number, and $f$ and $g$ are given functions. Note that the nonlinearity $f$ is not assumed to be subordinated to the Laplacian. The existence of a finite dimensional global attractor for system (1) is proved under some natural assumptions on the nonlinear term $f$.

In this paper we analyze the existence, the uniqueness and the stability of the positive steady-states of a sublinear logistic porous media equation involving a weight function vanishing on a region of the support domain. Quite surprisingly, the model possesses a unique positive steady state which attracts to all positive solutions within the range of values of the parameters for which the dynamics of the associated problem with linear diffusion is fully described by metasolutions. Metasolutions are extensions by infinity of large solutions in the support of the weight function. One of the main results of this paper shows how the classical solutions of the porous media equation approximate metasolutions as the nonlinear diffusion rate $m$ approaches one. The main technical difficulties that we must overcome in our analysis are the lost of a priori bounds in the vanishing region of the weight function and the fact that most of the underlying potentials are unbounded on the boundary of the support domain.

Using a generalized assumption of Osgood type, we prove a new comparison result for viscosity sub and supersolutions of fully nonlinear, possibly degenerate, parabolic equations. Our result allows to deal with hamiltonian functions with a quadratic growth in the spatial gradient, under special compatibility conditions with the diffusive terms. It applies in particular to a financial differential model for pricing Mortgage-Backed Securities.