We consider an initial and boundary value problem for a class of possibly degenerate second-order parabolic PDEs. The condition prescribed at the lateral boundary of the domain is of non local type, requiring the time derivative of the unknown to equal a given function of the total flux through the boundary itself (such conditions arise in problems involving contact with a body where diffusivity is infinite). We give results of local-in-time existence of (weak) solutions, and then we investigate the existence of solutions defined for all positive times. We show that, due to certain nonlinearities appearing in the problem, blow up in a finite time may occur, and we give, by means of suitable a~priori estimates, criteria for the global existence of solutions, or for occurrence of blow up.

We prove the existence of $\text{ cat } \Omega +1$ positive solutions of problem (1). If $\text{ cat } \Omega >1$ then we establish the existence of $\text{ cat } \Omega +2$ positive solutions. The proofs are based on the Lusternik-Schnirelman theory of critical points.

We establish here a weak stability result for sequences of solutions of the Landau equation. This result relies on the compactness properties recently shown to hold for the equation, and involves a defect measure. It enables us to prove the global existence of a renormalized solution of the ``Landau equation with a defect measure''. We establish an entropy inequality for this solution, and then study its long-time behavior.

This paper shows the existence of time-global continuous characteristic solutions of the Cauchy problem for a class of $2 \times 2$ weakly hyperbolic systems of conservation laws in one space dimension with possibly unbounded initial propagation speed, and gives a condition on the growth of the initial propagation speed necessary and sufficient for the uniqueness of continuous solutions.

We study the existence of positive solutions for the equation $\Delta u+ K(x) u^{\frac{n+2}{n-2}}=0$ in $\mathbb{R}^n$ ($n \geq 3$) which decay to $0$ at infinity like $|x|^{2-n}$; $K(x)$ is a function which is bounded from above and below by positive constants, and no symmetry assumption on $K$ is made. We find conditions that guarantee existence for a large class of $K$'s. As a consequence one can explicitly show that the set of coefficients for which a solution exists is dense, in $C^1$ norm, in the space of positive bounded $C^1$ functions. These conditions also allow us to display a radial $K$ such that the previous problem has nonradial solutions but no radial solution. Some new results for the corresponding problem on $\Sn$ are also proved.

In this paper we prove existence of a nontrivial solution for Hamiltonian systems of the form $$ \begin{cases} -\triangle u = \delta u + \gamma v + \frac{\partial H}{\partial v} (x, u, v) \\ -\triangle v = \lambda u + \delta v + \frac{\partial H}{\partial u} (x, u, v) & \mbox{in }\;\; \Omega, \end{cases} $$ subject to Dirichlet boundary conditions. The method used is variational through a generalized mountain pass theorem for indefinite functionals due to Benci-Rabinowitz in a version introduced by Felmer

In this paper we prove some existence and regularity results for equations of the type $- \text{div}\, (a(u,u,\Delta u)+ K(x,u)) +H(x,\Delta u) =f $, in a bounded open set $\Omega$, $u=0 $ on $ \partial \Omega$, where $a$, $K$ and $H$ are Caratheodory functions and $f$ is a Radon measure.