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Wednesday, August 21, 2019 - 13:00 in V2-213

Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations

A talk in the Bielefeld Stochastic Summer series by
Michael Röckner from Bielefeld

Abstract: In this talk we shall identify generalized time-fractional derivatives as generators of $C_0$-operator semigroups and prove their strong dissipativity on Gelfand triples of properly in time weighted $L^2$-path spaces. In particular, the classical Caputo derivative is included as a special case. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of type $$ \frac{d}{dt}(k * u)(t) + A(t, u(t)) = f(t), \quad t \in (0,T), $$ with (in general nonlinear) operators $A(t,\cdot)$ satisfying general weak monotonicity conditions. Here $k$ is a non-increasing locally Lebesgue-integrable nonnegative function on $[0, \infty)$ with $\lim\limits_{s\rightarrow\infty} k(s)=0$. Analogous results for the case, where $f$ is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators or the time-fractional (stochastic) $p$-Laplace equation are covered.

Joint work with: Wei Liu (Jiangsu Normal University, Xuzhou) and José Luís da Silva (University of Madeira, Funchal)

References: Liu/R./Silva: arXiv:1908.03959

Within the CRC this talk is associated to the project(s): A5, B1, B2


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