We investigate the singular behaviour of any solution of the semilinear parabolic equation $u_{t}-\Delta u+\left| u\right| ^{q-1}u=0$ in a cylindrical domain near a boundary point. The description is made by means of explicit self-similar solutions of the same equation into a half-space. Uniqueness of singular solutions is also proven.

This paper deals with a sublinear elliptic equation $\Delta u + f(u)=0$ in a ball with the Dirichlet condition $u=0$ on the boundary. Under a suitable assumption on $f(\cdot)$, we establish the necessary and sufficient condition for the existence and uniqueness of radially symmetric nodal solutions.

We study the variety of solutions of the inhomogeneous div--curl problem in exterior domains in dependence on the decay conditions on div and curl. Here we consider the Neumann as well as the Dirichlet boundary value prescription where in the first case the topological impact is decisive. In the second case the integrability conditions on div, curl and the boundary values are more difficult. Finally we present Hölder estimates for the solution of the Dirichlet or Neumann problem where it is unique.

Based on an analytic semigroup setting, we first consider semilinear reaction--diffusion equations with spatially quasiperiodic coefficients in the nonlinearity, rapidly varying on spatial scale $\varepsilon$. Under periodic boundary conditions, we derive quantitative homogenization estimates of order $\varepsilon^\gamma$ on strong Sobolev spaces $H^\sigma$ in the triangle $$0 <\gamma < \min (\sigma -n/2,2-\sigma).$$ Here $n$ denotes spatial dimension. The estimates measure the distance to a solution of the homogenized equation with the same initial condition, on bounded time intervals. The same estimates hold for $C^1$ convergence of local stable and unstable manifolds of hyperbolic equilibria. As a second example, we apply our abstract semigroup result to homogenization of the Navier--Stokes equations with spatially rapidly varying quasiperiodic forces in space dimensions 2 and 3. In both examples, a Diophantine condition on the spatial frequencies is crucial to our homogenization results. Our Diophantine condition is satisfied for sets of frequency vectors of full Lebesgue measure. In the companion paper [7], based on $L^2$ methods, these results are extended to quantitative homogenization of global attractors in near-gradient reaction--diffusion systems.