### 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 |
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Pages (from-to) | 418-447 |

Number of pages | 30 |

Journal | Differential and Integral Equations |

Volume | 196 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2004 |

### Keywords

- Wiskunde en Informatica
- Techniek
- technische Wiskunde en Informatica
- academic journal papers
- Peer-lijst tijdschrift

## Cite this

Clément, PPJE., Londen, SO., & Simonett, G. (2004). Quasilinear evolutionary equations and continuous interpolation spaces.

*Differential and Integral Equations*,*196*(2), 418-447. https://doi.org/10.1016/j.jde.2003.07.014