We study the electrostatic MEMS-device equation, \(u_{t} -\Delta u = \frac{\lambda |x|^{\beta}}{(1-u)^{p}}\), with Dirichlet boundary condition. First, we describe the touchdown of non-stationary solution in accordance with the total set of stationary solutions. Then, we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we show the Morse-Smale property for radially symmetric non-stationary solutions.

Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let $\Omega$ be a $C^2$ bounded region in ${\mathbf {R}}^n$ ( $n=2, 3$ ). The double layer potential $K: L^2(\partial \Omega ) \rightarrow L^2(\partial \Omega )$ is defined by \[ $(K \psi )(x) \equiv \int _{\partial \Omega } \psi (y)\cdot \nu _{y} E(x, y) \; ds_y,$ \] where \[ $E(x, y)= \begin {cases} \frac {1}{\pi } \log \frac {1}{|x-y|}, \quad \;\mbox {if}\; n=2, \\ \frac {1}{2\pi } \frac {1}{|x-y|}, \quad \hspace {4mm}\;\mbox {if} \; n=3, \end {cases}$ \] $ds_y$ is the line or surface element and $\nu _y$ is the outer normal derivative on $\partial \Omega$ . It is known that $K$ is a compact operator on $L^2(\partial \Omega )$ and consists of at most a countable number of eigenvalues, with $0$ as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of $\partial \Omega$ .

We consider a system which describes the scaling limit of several chemotaxis systems. We focus on self-similarity, and review some recent results on forward and backward self-similar solutions to the system.

This paper considers quadratic and super-quadratic reaction-diffusion systems, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.