In this paper diﬀusion processes with changing modes are studied involving the variable order partial diﬀerential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-diﬀerential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diﬀusion modes change. These memory eﬀects are classiﬁed and studied in detail. Processes that have distinctive regimes of diﬀerent types of diﬀusion depending on time are ubiquitous in the nature. Examples include diﬀusion in a heterogeneous media and protein movement in cell biology.
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Sabir Umarov, Stanly Steinberg, Variable Order Diﬀerential Equations with Piecewise Constant Order-Function and Diﬀusion with Changing Modes. Z. Anal. Anwend. 28 (2009), no. 4, pp. 431–450DOI 10.4171/ZAA/1392