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

Abstract

For $\nu ,{\nu }_i,{\mu }_j\in (0,1)$, we analyze the semilinear integro-differential equation on the one-dimensional domain $\Omega =(a,b)$ in the unknown $u=u(x,t)$ \[ \mathbf{D}_{t}^ν(\varrho_{0}u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{ν_{i}}(\varrho_{i}u) -\sum_{j=1}^{N}\mathbf{D}_{t}^{μ_{j}}(γ_{j}u) -\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u)=g(x,t), \] 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.

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