Linear Parabolic Equations in Locally Uniform Spaces

Abstract

We analyze the linear theory of parabolic equations in uniform spaces. We obtain sharp Lp-Lq-type estimates in uniform spaces for heat and Schrödinger semigroups and analyze the regularizing effect and the exponential type of these semigroups. We also deal with general second-order elliptic operators and study the generation of analytic semigoups in uniform spaces.

Dissipative parabolic equations in locally uniform spaces

“…Finally in the Appendix we will give suitable characterization of the spacesẆ s,p U R N which are used in previous sections. Such a result will supplement earlier consideration of [7].…”

“…More recently Bessel potentials spaces H s p R N have been used in [6]. The so-called locally uniform spaces have been used in [12,13,33,34] and a systematic study of linear equations in such spaces was developed in [7]. Hence this paper is a natural "nonlinear" continuation of that one.…”

“…is defined by interpolation, see [7] and the Appendix for details. We just note here that translations are continuous in these spaces.…”

“…[32]. On the other hand for the case of weighted spaces, although the linear semigroup is continuous (see [7]), the difficulties connected with the nonlinear equations come from the lack of Sobolev-like embeddings which make unavailable the standard techniques for handling nonlinear terms. In fact it was shown in [8] that generally, without specific structure assumption on nonlinear terms, semilinear problems are ill posed in weighted spaces when the weight goes to zero at infinity.…”

“…In this section we recall some results reported in [7] that have been used before in this paper and prove an embedding property that has been used in the proof of Lemma 2. p (B(y,1) …”