Existence and uniqueness of weak and strong solutions for semilinear differential and integrodifferential abstract equations with a Hille-Yosida operator are proved together with criteria for the global existence of the solutions. An application is given to a semilinear one-dimensional wave equation.

In this paper, we provide sufficient conditions for the existence of a unique mild solution on a semi-infinite interval for two classes of first-order partial functional and neutral functional differential evolution equations with infinite delay using a recent nonlinear alternative of Leray Schauder type due to Frigon and Granas for contractions maps in Fréchet spaces, combined with semigroup theory.

A Dirichlet problem for orthogonal Hessians in two dimensions is explicitly solved, by characterizing all piecewise $C^{2}$ functions $u:\Omega \subset\mathbb{R}^{2}\rightarrow\mathbb{R}$ with orthogonal Hessian in terms of a property named 'second order angle condition>" as in (1.1).

When particle speeds are large the motion of a collisionless plasma is modeled by the relativistic Vlasov Maxwell system. Large time behavior of solutions which depend on one position variable and two momentum variables is considered. In the case of a single species of charge it is shown that there are solutions for which the charge density $(\rho = \int f dv)$ does not decay in time. This is in marked contrast to results for the non-relativistic Vlasov Poisson system in one space dimension. The case when two oppositely charged species are present and the net total charge is zero is also considered. In this case, it is shown that the support in the first component of momentum can grow at most as $t^{\frac{3}{4}}$.

This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting in both equations. We will prove that the solution of our considered system blows up in finite time provided that the initial data are large enough. This result extends a previous result in [1] to a large class of initial data. The key ingredient in the proof is a method used in [25] with necessary modifications imposed by the nature of our problem.

In this paper we continue our study, started in [9], on the regularity theory of Stefan-like free boundary problems for a special class of fully nonlinear equations of parabolic type. We prove that degenerate Lipschitz free boundaries, with small Lipschitz constant in space, are $C^1$.

Using variational methods we establish the existence of solutions for the following class of $p(x)$-Laplacian equations: \begin{equation} -div(|\nabla u|^{p(x)-2}\nabla u) + |u|^{p(x)-2}u= \lambda |u|^{q(x)-2}u + |u|^{p^{*}(x)-2}u, \,\,\,\,\, \mathbb{R}^{N}, \tag*{(P)} \end{equation} where $\lambda \in (0, \infty)$ is a parameter and $p(x), q(x): \mathbb{R}^{N} \to \mathbb{R}$ are radial continuous functions satisfying $1 < p(x) < N$ and $p(x) < < q(x) < < p^{*}(x)= \frac{p(x)N}{N-p(x)}$.

This paper is concerned with nonlinear and spectral stability of periodic traveling wave solutions for a nonlinear Schrödinger type system arising in nonlinear optics. We prove the existence of two smooth curves of periodic solutions depending on the cnoidal type functions. In the framework established by Grillakis, Shatah and Strauss we prove a stability result under perturbations having the same minimal wavelength and zero mean over their fundamental period. By using the so-called Bloch wave decomposition theory we show spectral stability for a general class of periodic solutions.

We investigate a non-local, non-linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see [6, 7]. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well posedness for initial data in $L^2$ and give an explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results.

We consider a stochastic functional differential equation with an arbitrary Lipschitz diffusion coefficient depending on the past. The drift part contains a term with superlinear growth and satisfying a dissipativity condition. We prove tightness and Feller property of the segment process to show existence of an invariant measure.