We consider a nonlinear parabolic equation involving nonmonotone diffusion. Existence and uniqueness of solutions are obtained, employing methods for semibounded evolution equations. Also shown is the existence of a global attractor for the corresponding dynamical system.

We study $H$-systems with a Dirichlet boundary data $g$. Under some conditions, we show that if the problem admits a solution for some $\left(H_0,g_0\right)$, then it can be solved for any $\left(H,g\right)$ close enough to $\left(H_0,g_0\right)$. Moreover, we construct a solution of the problem applying a Newton iteration.

A bound sets technique is developed for Floquet problems of Carathéodory differential inclusions. It relies on the construction of either continuous or locally Lipschitzian Lyapunov-like bounding functions. Proceeding sequentially, the existence of bounded trajectories is then obtained. Nontrivial examples are supplied to illustrate our approach.