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

Abstract

For ν, ν i , µ j ∈ (0, 1), we analyze the semilinear integro-differential equation on the onedimensional domain Ω = (a, b) in the unknown u = u(x, t)where, L k are uniform elliptic operators with time-dependent smooth coefficients, 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 ν, ν i , µ 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.