A P ] 2 3 A ug 2 01 9 STRONG DISSIPATIVITY OF GENERALIZED TIME-FRACTIONAL DERIVATIVES AND QUASI-LINEAR ( STOCHASTIC ) PARTIAL DIFFERENTIAL EQUATIONS

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted L-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 d dt (k ∗ u)(t) +A(t, u(t)) = f(t), 0 < t < T, with (in general nonlinear) operators A(t, ·) satisfying general weak monotonicity conditions. Here k is a non-increasing locally Lebesgue-integrable nonnegative function on [0,∞) with lim s→∞ 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 quasilinear (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.

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