We establish the existence of a unique solution of an initial boundary value problem for the nonstationary Stokes equations in a bounded fixed cylindrical domain with measure data. Feedback laws yield the source and its intensity from the partial measurements of the solution in a subdomain.

We give necessary and sufficient conditions for the existence of positive solutions for sublinear Dirichlet periodic parabolic problems $Lu=g\left(x,t,u\right)$ in $\Omega \times ℝ$ (where $\Omega \subset {ℝ}^N$ is a smooth bounded domain) for a wide class of Carathéodory functions $g:\Omega \times ℝ\times [0,\infty )\to ℝ$ satisfying some integrability and positivity conditions.

We obtain the zero distribution of sequences of classical orthogonal polynomials associated with Jacobi, Laguerre, and Hermite weights. We show that the limit measure is the extremal measure associated with the corresponding weight.

We consider the one-dimensional logistic problem ${\left(r^{\alpha }A\left(\left|u^{\prime }\right|\right)u^{\prime }\right)}^{\prime }=r^{\alpha }p\left(r\right)f\left(u\right)$ on $\left(0,\infty \right)$, $u\left(0\right)>0$, $u^{\prime }\left(0\right)=0$, where $\alpha$ is a positive constant and $A$ is a continuous function such that the mapping $tA\left(\left|t\right|\right)$ is increasing on $\left(0,\infty \right)$. The framework includes the case where $f$ and $p$ are continuous and positive on $\left(0,\infty \right)$, $f\left(0\right)=0$, and $f$ is nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results depending on the growth of $p$ and $A$. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.