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 |