Semilinear subdiffusion with memory in the one-dimensional case

For ν∈(0,1), we consider the semilinear integro-differential equation on the one-dimensional domain Ω=(a,b) in the unknown u=u(x,t)Dtνu−L1u−∫0tK(t−s)L2u(⋅,s)ds=f(x,t,u)+g(x,t)where Dtν is the Caputo fractional derivative and L1 and L2 are uniform elliptic operators with time-dependent smooth coefficients. Under certain structural conditions on the nonlinearity f, the global existence and uniqueness of classical solutions to the related initial-boundary value problems are established, via the so-called continuation argument approach. The key point is looking for suitable a priori estimates of the solution in the fractional Hölder spaces.