We study the Dirichlet problem in the unit ball $B_1$ of $\mathbb R^2$ for the Hénon-type equation of the form \begin{equation*} \begin{cases} -\Delta u =\lambda u+|x|^{\alpha}f(u) & \mbox{in } B_1, \\ \quad \ \ u = 0 & \mbox{on } \partial B_1, \end{cases} \end{equation*} where $f(t)$ is a $C^1$-function in the critical growth range motivated by the celebrated Trudinger-Moser inequality. Under suitable hypotheses on constant $\lambda$ and $f(t)$, by variational methods, we study the solvability of this problem in appropriate Sobolev s paces.

This paper concerns a nonlinear viscoelastic wave equation with time-dependent delay. Under suitable relation between the weight of the delay and the weight of the term without delay, we prove the global existence of weak solutions by the combination of the Galerkin method and potential well theory. In addition, by assuming the minimal conditions on the $L^1(0,\infty)$ relaxation function $g$, namely, $g'(t)\leq-\xi(t)H(g(t))$, where $H$ is an increasing and convex function and $\xi$ is a nonincreasing differentiable function, and by using some properties of convex functions, we establish optimal explicit and general energy decay results. This result is new and substantially improves existing results in the literature.

Using operator semigroup methods, we show that Fokker-Planck type second-order PDEs can be used to approximate the evolution of the distribution of a one-step process on $N$ particles governed by a large system of ODEs. The error bound is shown to be of order $O(1/N)$, surpassing earlier results that yielded this order for the error only for the expected value of the process through mean-field approximations. We also present some conjectures showing that the methods used have the potential to yield even stronger bounds, up to $O(1/N^3)$.