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}^{\nu}(\varrho_{0}u)+\sum\limits_{i = 1}^{M}{\mathbf{D}}_{t}^{\nu_{i}}(\varrho_{i}u) -\sum\limits_{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 Hölder and Sobolev spaces. The problems are also studied from the numerical point of view.