##
**Fractional evolution equations in Banach spaces.**
*(English)*
Zbl 0989.34002

Eindhoven: Eindhoven University Press, Eindhoven: Eindhoven Univ. of Technology (Thesis), iv, 107 p. (2001).

This book is concerned with (strong) solvability as well as regularity, and related questions, for evolutionary equations with fractional time-derivative:
\[
D_t^\alpha \bigl(u(t)-u_0\bigr)= Au(t),\;t>0,
\]
with initial values \(u_0\) and a closed linear operator \(A\) that is densely defined on some Banach space \(X\). In particular, \(\alpha\in(0,2]\) is considered. The case \(\alpha=1\) corresponds to abstract parabolic problems describing diffusion processes – the solution operator forms a \(C_0\)-semigroup-, whereas \(\alpha=2\) is related to wave phenomena, and the solution operator forms a cosine operator family.

The fractional time derivative may be defined via the integral formulation \[ D^\alpha_t u(t):= {d^m\over dt^m}\int^t_0 {(t-s)^{m-1-\alpha} u(s) \over\Gamma (m-\alpha)}ds,\;m-1< \alpha\leq m,\;m\in\mathbb{N}. \] The main problem in the analysis of such problems seems to be the lack of the semigroup property or cosine functional equation as the fractional derivative represents to some extent a memory property and has a nonlocal character. The analysis employed by the author relies essentially on the theory of Volterra integral equations.

After the Introduction (4 pp.), some preliminaries, and setting up the problem in Chapter 1 (14 pp.), the author proves in Chapter 2 (19 pp.) generation, approximation, and perturbation results that extend the well-known Hille-Yosida theory. This is then followed in Chapter 3 (13 pp.) by a discussion of a subordination principle: the smaller \(\alpha\) the better the solution is. Regularity in \(L^p(X)\) and – if \(A\) is accretive and \(X\) is a Hilbert space – in the Hilbert space \(L^2(X)\) is analyzed in Chapters 4 and 5 (17 and 15 pp.), respectively, even for the nonhomogeneous equation with some right-hand side \(f\). The final Chapter 6 (11 pp.) is devoted to the global solvability for quasilinear equations with \(\alpha\in (1,2)\).

The list of references counts 75 items, most of them being research articles. The short index, which also lists symbols used for function spaces etc., helps the reader. Although the topic is quite special and the presentation, as typical for PhD theses, is rather dense and sometimes heavy, the book is clearly and well written. What the reader is maybe missing is some key to applications that are only named in the Preface.

The fractional time derivative may be defined via the integral formulation \[ D^\alpha_t u(t):= {d^m\over dt^m}\int^t_0 {(t-s)^{m-1-\alpha} u(s) \over\Gamma (m-\alpha)}ds,\;m-1< \alpha\leq m,\;m\in\mathbb{N}. \] The main problem in the analysis of such problems seems to be the lack of the semigroup property or cosine functional equation as the fractional derivative represents to some extent a memory property and has a nonlocal character. The analysis employed by the author relies essentially on the theory of Volterra integral equations.

After the Introduction (4 pp.), some preliminaries, and setting up the problem in Chapter 1 (14 pp.), the author proves in Chapter 2 (19 pp.) generation, approximation, and perturbation results that extend the well-known Hille-Yosida theory. This is then followed in Chapter 3 (13 pp.) by a discussion of a subordination principle: the smaller \(\alpha\) the better the solution is. Regularity in \(L^p(X)\) and – if \(A\) is accretive and \(X\) is a Hilbert space – in the Hilbert space \(L^2(X)\) is analyzed in Chapters 4 and 5 (17 and 15 pp.), respectively, even for the nonhomogeneous equation with some right-hand side \(f\). The final Chapter 6 (11 pp.) is devoted to the global solvability for quasilinear equations with \(\alpha\in (1,2)\).

The list of references counts 75 items, most of them being research articles. The short index, which also lists symbols used for function spaces etc., helps the reader. Although the topic is quite special and the presentation, as typical for PhD theses, is rather dense and sometimes heavy, the book is clearly and well written. What the reader is maybe missing is some key to applications that are only named in the Preface.

Reviewer: Etienne Emmrich (Berlin)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34G20 | Nonlinear differential equations in abstract spaces |

26A33 | Fractional derivatives and integrals |

35K90 | Abstract parabolic equations |

35L90 | Abstract hyperbolic equations |

47D06 | One-parameter semigroups and linear evolution equations |

47D09 | Operator sine and cosine functions and higher-order Cauchy problems |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |