Solvability of linear boundary value problems for subdiffusion equations with memory. (English) Zbl 1402.35304

Krasnoschok, Mykola; Pata, Vittorino; Vasylyeva, Nataliya

J. Integral Equations Appl. 30, No. 3, 417-445 (2018).

Summary: For \(\nu \in (0,1)\), the nonautonomous integro-differential equation \[ \mathbf {D}_{t}^{\nu }u-\mathcal {L}_{1}u-\int _{0}^{t}\mathcal {K}_{1}(t-s)\mathcal {L}_{2}u(\cdot ,s)\,ds =f(x,t) \] is considered here, where \(\mathbf {D}_{t}^{\nu }\) is the Caputo fractional derivative and \(\mathcal {L}_{1}\) and \(\mathcal {L}_{2}\) are uniformly elliptic operators with smooth coefficients dependent on time. The global classical solvability of the associated initial-boundary value problems is addressed.

MSC:

35R11	Fractional partial differential equations
35C15	Integral representations of solutions to PDEs
45N05	Abstract integral equations, integral equations in abstract spaces

Keywords:

materials with memory; subdiffusion equations; Caputo derivatives; coercive estimates

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