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

Abstract

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted $L^p$-path spaces is proved. 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 \begin{equation*} \frac{d}{dt} (k * u)(t) + A(t, u(t)) = f(t), \quad 0<t<T, \end{equation*} 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 $\underset{s\to \infty }{\lim }k(s)=0$. 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.

34 pages. Some typos are corrected and some references are added in the new version. arXiv admin note: text overlap with arXiv:1708.05649