Smoothing properties of heat semigroups in infinite dimensions

Abstract

Let us consider first a finite-dimensional problem

$$ \left\{ {\begin{array}{*{20}{c}} {{u_{t}}\left( {t,x} \right) = \frac{1}{2}\sum\limits_{{k = 1}}^{n} {{\lambda _{k}}\frac{{{\partial ^{2}}u\left( {t,x} \right)}}{{\partial x_{k}^{2}}}} t\underline > 0} \\ {u\left( {0,x} \right) = \varphi \left( x \right),x = \left( {{x_{1}}, \ldots ,{x_{n}}} \right)} \\ \end{array} } \right. $$

(1)

where φ∈C _b(Rⁿ), the space of all uniformly continuous and bounded mappings Rⁿ→R and λ₁,…,λ_n are positive numbers.

Partially supported by the Italian National Project MURST ”Problemi nonlineari nell’Analisi …”