In this paper we survey some recent results on parabolic equations on curved squeezed domains. More specifically, consider the family of semilinear Neumann boundary value problems $$ \alignedat2 u_t &= \Delta u + f(u), &\quad&t> 0,\ x\in \Omega_\varepsilon, \\ \partial_{\nu_\varepsilon}u&= 0, &\quad& t> 0,\ x\in \partial \Omega_\varepsilon \endaligned \leqno{(\text{\rm E}_\varepsilon)} $$ where, for $\varepsilon> 0$ small, the set $\Omega_\varepsilon$ is a thin domain in $\mathbb R^\ell$, possibly with holes, which collapses, as $\varepsilon\to0^+$, onto a (curved) $k$-dimensional submanifold $\mathcal M$ of $\mathbb R^\ell$. If $f$ is dissipative, then equation (E$_\varepsilon$) has a global attractor ${\mathcal A}_\varepsilon$. We identify a "limit" equation for the family (E$_\varepsilon$), establish an upper semicontinuity result for the family ${\mathcal A}_\varepsilon$ and prove an inertial manifold theorem in case $\mathcal M$ is a $k$-sphere.

We discuss the calculation of critical groups for jumping nonlinearities as the resonance set is crossed. In addition, we produce a counter-example showing that even "generically" the resonance set is more complicated than previously thought.

For a ramified covering between Riemann surfaces the groups Deck of deck transformations and Mon of monodromy permutations are introduced. We associate with them groups of automorphisms of certain extensions of function fields. We study relations between these objects. \endabstract

We consider homogeneous polynomial maps $F\colon {\mathbb R}^{n} \rightarrow {\mathbb R}^{n}$ of degree $p$. We classify the pairs $(p,n)$ for which there exists a surjective and non-proper such map and when the right inverse to $F$ exists but is unbounded.

In this article we discuss the existence and non-existence of forced $T$-periodic solutions to ordinary differential equations of the form $u''+g(u)=e(t)$. The results concern equations with bounded nonlinear terms $g$ satisfying $g(s)> 0$ (or $g(s)< 0$) for all real numbers $s$, and $g(\pm \infty )=0$. Variational and topological methods are employed.

Energy-transport models introduced by R. F. Streater describe the evolution of the density and temperature of a cloud of self-gravitating particles. We study the existence of steady states with prescribed mass and energy for these models.

We consider the boundary value problem \begin{gather*} u^{(4)}(x) =g(u(x)) + p(x,u^{(0)}(x),\dots,u^{(3)}(x)) , \quad x \in (0,1), \\ u(0) =u(1) =u^{(b)}(0) =u^{(b)}(1) =0, \end{gather*} where:

(i) $g \colon \mathbb R \to \mathbb R$ is continuous and satisfies $\lim_{|\xi| \to \infty} g(\xi)/\xi =\infty$ ($g$ is superlinear as $|\xi| \to \infty$),

(ii) $p \colon [0,1] \times \mathbb R^4 \to \mathbb R$ is continuous and satisfies $$ |p(x,\xi_0,\xi_1,\xi_2,\xi_3)| \le C + \frac{1}{4} |\xi_0| , \quad x \in [0,1],\ (\xi_0,\xi_1,\xi_2,\xi_3) \in \mathbb R^4, $$ for some $C> 0$,

(iii) either $b=1$ or $b=2$.

We obtain solutions having specified nodal properties. In particular, the problem has infinitely many solutions.

In the paper the equations describing the motion of a drop of a viscous heat-conducting capillary fluid bounded by a free surface are examined. Assuming that the viscosity coefficients, the coefficient of heat-conductivity, the pressue and the specific heat at constant volume of the fluid depend on its density and temperature we prove the existence of a global in time solution which is close to a constant state for any moment of time.

The Lefschetz fixed point theorem for multivalued upper semi-continuous acyclic maps with multiplicity with respect to (w.r.t.) a given field $\mathbb F$ of $\mathbb F$-simplicial spaces is proved.

An upper and lower solution approach is presented for singular boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.

Let $(\Omega ,\Sigma )$ be a measurable space and $X$ be a separable metric space. It is shown that for measurable maps $\zeta ,\eta \colon \Omega \rightarrow X$, if a random map $T\colon\Omega \times X\rightarrow X$ satisfies $d(T(\omega ,\zeta (\omega )),T(\omega ,\eta (\omega )))\leq \alpha d(\zeta (\omega ),\eta (\omega ))+\gamma $ then $\inf\{d(\xi (\omega ),T(\omega ,\xi (\omega )))\}\leq \gamma/ (1-\alpha)$, where $\gamma > 0$, $\alpha \in (0,1)$ and $\inf$ is taken over all measurable maps $\xi \colon \Omega \rightarrow X$. Several consequences of this result are also obtained.