INITIAL-BOUNDARY VALUE PROBLEMS TO SEMILINEAR MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS

For nu, nu(i), mu(j) is an element of (0, 1), we analyze the semilinear integro-differential equation on the one-dimensional domain Omega = (a, b) in the unknown u = u(x, t) D-t(nu) (rho(0)u) + Sigma D-M(i=1)t(nu i) (rho(i)u) - Sigma D-N(j=1)t(mu j) (gamma ju) - L(1)u - kappa * L(2)u + f(u) = g(x, t), where D-t(nu), D-t(nu i), D-t(mu j) are Caputo fractional derivatives, rho(0) = rho(0)(t) > 0, rho(i) = rho(i)(t), gamma(j) = gamma(j)(t), L-k are uniform elliptic operators with time-dependent smooth coefficients, kappa 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 Holder and Sobolev spaces. The problems are also studied from the numerical point of view.