  For $0<\nu_2<\nu_1\leq 1$, we analyze a linear integro-differential equation
on the space-time cylinder $\Omega\times(0,T)$ in the unknown $u=u(x,t)$
$$\mathbf{D}_{t}^{\nu_1}(\varrho_{1}u)-\mathbf{D}_{t}^{\nu_2}(\varrho_2
u)-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u =f$$ where
$\mathbf{D}_{t}^{\nu_i}$ are the Caputo fractional derivatives,
$\varrho_i=\varrho_i(x,t)$ with $\varrho_1\geq \mu_0>0$, $\mathcal{L}_{i}$ are
uniform elliptic operators with time-dependent smooth coefficients,
$\mathcal{K}$ 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.
