English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study heat and wave type equations on a separable Hilbert space$\mathcal{H}$ by considering non-local operators in time with any positivedensely defined linear operator with discrete spectrum. We show the explicitrepresentation of the solution and analyse the time-decay rate in a scale ofsuitable Sobolev space. We perform similar analysis on multi-term heat andmulti-wave type equations. The main tool here is the Fourier analysis which canbe developed in a separable Hilbert space based on the linear operatorinvolved. As an application, the same Cauchy problems are considered andanalysed in the setting of a graded Lie group. In this case our analysis relieson the group Fourier analysis. An extra ingredient in this framework allows, inthe case of heat type equations, to establish $L^p$-$L^q$ estimates for$1\leqslant p\leqslant 2\leqslant q<+\infty$ for the solutions on graded Liegroup groups. Examples and applications of the developed theory are given,either in terms of self-adjoint operators on compact or non-compact manifolds,or in the case of particular settings of graded Lie groups. The results of thispaper significantly extend in different directions the results of Part I, whereoperators on compact Lie groups were considered. We note that the resultsobtained in this paper are also new already in the Euclidean setting of$\mathbb{R}^n$.

Heat and wave type equations with non-local operators, II. Hilbert spaces and graded Lie groups