We prove a priori estimates for solutions of the Poisson and heat equations in weighted spaces of Kondrat'ev type. The weight is a power of the distance from a distinguished axis.

In this survey we consider superlinear parabolic problems which possess both blowing-up and global solutions and we study a priori estimates of global solutions.

Let $A$ be a closed set of $M\simeq {\mathbb{R}}^n$, whose conormai cones $x+y_x^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a $C^1$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^n$, using cohomology groups of ${\mathcal{O}}_X$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi }}^{-1}\left(x_0\right)}$, $x_0\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi }_i^C\right)\right\}}_{i=1,\mathrm{\dots },m}$, $\left\{{\varphi }_i\right\}$ being a base for the linear hull of ${\gamma }_{x_0}^{*}\left(A\right)$ in $T_{x_0}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma }}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_M\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with $C^{\omega }$-boundary,...

Let T be a semigroup of linear contractions on a Banach space X, and let $X_s\left(T\right)=x\in X:li{m}_{s\to \infty }\parallel T\left(s\right)x\parallel =0$. Then $X_s\left(T\right)$ is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then $X_s\left(T\right)$ is the annihilator of the unitary eigenvectors of T*, and $li{m}_s\parallel T\left(s\right)x\parallel =inf\parallel x-y\parallel :y\in X_s\left(T\right)$ for each x in X.