Initial-boundary value problems to semilinear multi-term fractional differential equations

For $\nu ,{\nu }_i,{\mu }_j\in (0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $\mathrm{\Omega }=(a,b)$ in the unknown $u=u(x,t)$ $${\mathbf{D}}_t^{\nu }({\varrho }_{0}u)+\sum _{i=1}^M{\mathbf{D}}_t^{{\nu }_i}({\varrho }_{i}u)-\sum _{j=1}^N{\mathbf{D}}_t^{{\mu }_j}({\gamma }_{j}u)-{\mathcal{L}}_{1}u-\mathcal{K}\ast {\mathcal{L}}_{2}u+f(u)=g(x,t),$$ where ${\mathbf{D}}_t^{\nu },{\mathbf{D}}_t^{{\nu }_i},{\mathbf{D}}_t^{{\mu }_j}$ are Caputo fractional derivatives, ${\varrho }_0={\varrho }_0(t)>0,$ ${\varrho }_i={\varrho }_i(t)$, ${\gamma }_j={\gamma }_j(t)$, ${\mathcal{L}}_k$ are uniform elliptic operators with time-dependent smooth coefficients, $\mathcal{K}$ is a summable convolution kernel. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under certain structural conditions on the nonlinearity $f$ and orders $\nu ,{\nu }_i,{\mu }_j$, the global existence and uniqueness of classical and strong solutions to the related initial-boundary value problems are established via the so-called continuation arguments method. The crucial point is searching suitable a priori estimates of the solution in the fractional Hölder and Sobolev spaces. The problems are also studied from the numerical point of view.