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

TL;DR

This paper studies initial-boundary value problems for complex semilinear multi-term fractional differential equations, establishing existence, uniqueness, and numerical analysis under certain conditions, with applications to biological models like oxygen transport.

Contribution

It introduces new analytical techniques for proving existence and uniqueness of solutions to multi-term fractional PDEs with variable coefficients and convolution terms.

Findings

Proved global existence and uniqueness of solutions.

Developed a priori estimates in fractional Sobolev spaces.

Explored numerical approaches for the 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 $u=u(x,t)$ \[ \mathbf{D}_{t}^{\nu}(\varrho_{0}u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{\nu_{i}}(\varrho_{i}u) -\sum_{j=1}^{N}\mathbf{D}_{t}^{\mu_{j}}(\gamma_{j}u) -\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u)=g(x,t), \] 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 H\"{o}lder and Sobolev spaces. The problems are also studied from the numerical point of view.