Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries

10/10/2022

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by   Vittorino Pata, et al.

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For 0<ν_2<ν_1≤ 1, we analyze a linear integro-differential equation on the space-time cylinder Ω×(0,T) in the unknown u=u(x,t) 𝐃_t^ν_1(ϱ_1u)-𝐃_t^ν_2(ϱ_2 u)-ℒ_1u-𝒦*ℒ_2u =f where 𝐃_t^ν_i are the Caputo fractional derivatives, ϱ_i=ϱ_i(x,t) with ϱ_1≥μ_0>0, ℒ_i are uniform elliptic operators with time-dependent smooth coefficients, 𝒦 is a summable convolution kernel, and f is an external force. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under suitable conditions on the given data, the global classical solvability of the associated initial-boundary value problems is addressed. To this end, a special technique is needed, adapting the concept of a regularizer from the theory of parabolic equations. This allows us to remove the usual assumption about the nonnegativity of the kernel representing fractional derivatives. The problem is also investigated from the numerical point of view.