Consider the nonlinear wave equation with zero mass in two space dimensions. When it comes to the associated Cauchy problem with small initial data, the known existence results are already sharp; those require the data to decay at a rate $k\geq k_c$, where $k_c$ is a critical decay rate that depends on the order of the nonlinearity. However, the known scattering results treat only the supercritical case $k>k_c$. In this paper, we prove the existence of the scattering operator for the full optimal range $k\geq k_c$.

In this paper, we establish a quantitative unique continuation property for some time-periodic linear parabolic equations in a bounded domain $\Omega$. We prove that for a time-periodic heat equation with particular time-periodic potential, its solution $u(x,t)$ satisfies $\left\| u(\cdot,0) \right\| _{L^{2}(\Omega) }\leq C\left\| u(\cdot,0) \right\| _{L^{2}(\omega) }$ where $\omega\subset\Omega$. Also we deduce the asymptotic controllability for the heat equation with an even, time-periodic potential. Moreover, the controller belongs to a finite dimensional subspace and is explicitly computed.

We study the existence of multiple positive solutions to systems of the form \begin{equation*} \begin{cases} \qquad-{\Delta} u ={\lambda} f(v), & \text{ in }{\Omega},\\ \qquad-{\Delta} v ={\lambda} g(u), & \text{ in }{\Omega},\\ \qquad\quad~~ u=0=v, & \text{ on }{\partial}{\Omega}. \end{cases} \end{equation*} Here ${\Delta}$ is the Laplacian operator, ${\lambda}$ is a positive parameter, ${\Omega}$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary and $f, g$ belong to a class of positive functions that have a combined sublinear effect at $\infty$. Our results also easily extend to the corresponding p-Laplacian systems. We prove our results by the method of sub and super solutions.

We prove that the solutions of the kinetic equation modelling the reflection of particles according to a reversible law and considered on the $n$-torus converge to the diffusion equation when the main free path goes to zero. This extends the work of Bardos, Golse and Colonna [2] to the case of any $n$-dimensional torus ergodic automorphism.

In this paper we present a modification of the usual {Proteus mirabilis Swarm model. For the obtained model (which is a two phase model with a non-linear diffusion term containing memory) we set up a collection of a priori estimates. Those estimates allow us to get an existence and uniqueness result.