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Diffusion equations with general nonlocal time and space derivatives. (English) Zbl 1443.60076

Summary: In the present study, firstly, based on the continuous time random walk (CTRW) theory, general diffusion equations are derived. The time derivative is taken as the general Caputo-type derivative introduced by A. N. Kochubei [Integral Equations Oper. Theory 71, No. 4, 583–600 (2011; Zbl 1250.26006)] and the spatial derivative is the general Laplacian defined by removing the conditions (1.5) and (1.6) from the definition of the general fractional Laplacian proposed in the paper [R. Servadei and E. Valdinoci, J. Math. Anal. Appl. 389, No. 2, 887–898 (2012; Zbl 1234.35291)]. Secondly, the existence of solutions of the Cauchy problem for the general diffusion equation is proved by extending the domain of the general Laplacian to a general Sobolev space. The results for positivity and boundedness of the solutions are also obtained. In the last, the existence result for solutions of the initial boundary value problem (IBVP) for the general diffusion equation on a bounded domain is established by using the Friedrichs extension of the general fractional Laplacian introduced in the book [G. Molica Bisci et al., Variational methods for nonlocal fractional problems. Cambridge: Cambridge University Press (2016; Zbl 1356.49003)].

MSC:

60J60 Diffusion processes
35R11 Fractional partial differential equations
60G51 Processes with independent increments; Lévy processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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