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Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. (English) Zbl 1189.35360
Summary: Some uniqueness and existence results for the solutions of the initial-boundary-value problems for the generalized time-fractional diffusion equation over an open bounded domain $G×\left(0,T\right)$, $G\subset {ℝ}^{n}$ are given. To establish the uniqueness of the solution, a maximum principle for the generalized time-fractional diffusion equation is used. In turn, the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is considered in the paper, too. Another important consequence of the maximum principle is the continuous dependence of the solution on the problem data. To show the existence of the solution, the Fourier method of the variable separation is used to construct a formal solution. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional diffusion equation that turns out to be a classical solution under some additional conditions.
##### MSC:
 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals (real functions) 35A01 Existence problems for PDE: global existence, local existence, non-existence 35A02 Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness
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