# Quasilinear evolutionary equations and continuous interpolation spaces

PPJE Clément, SO Londen, G Simonett

Research output: Contribution to journalArticleScientificpeer-review

53 Citations (Scopus)

## Abstract

Authors' abstract: In this paper we analyze the abstract parabolic evolutionary equations $$D_t^\alpha(u-x)+A(u)u=f(u)+h(t),\quad u(0)=x,$$ in continuous interpolation spaces allowing a singularity as $t\downarrow 0$. Here $D_t^\alpha$ denotes the time-derivative of order $\alpha\in(0,2)$. We first give a treatment of fractional derivatives in the spaces $L^p((0,T);X)$ and then consider these derivatives in spaces of continuous functions having (at most) a prescribed singularity as $t\downarrow 0$. The corresponding trace spaces are characterized and the dependence on $\alpha$ is demonstrated. Via maximal regularity results on the linear equation $$D_t^\alpha(u-x)+Au=f,\quad u(0)=x,$$ we arrive at results on existence, uniqueness and continuation on the quasilinear equation. Finally, an example is presented. [ Irina V.Melnikova (Ekaterinburg) ]
Original language Undefined/Unknown 418-447 30 Differential and Integral Equations 196 2 https://doi.org/10.1016/j.jde.2003.07.014 Published - 2004

### Bibliographical note

niet eerder opgevoerd sb

## Keywords

• Wiskunde en Informatica
• Techniek
• technische Wiskunde en Informatica