Stability of Runge-Kutta methods for abstract time-dependent parabolic problems: The Hölder case

We consider an abstract time-dependent, linear parabolic problem u′(t) = A(t)u(t), u(t0) = u0, where A(t) : D ⊂ X → X, t ∈ J , is a family of sectorial operators in a Banach space X with time-independent domain D. This problem is discretized in time by means of an A(θ) strongly stable Runge-Kutta method, 0 < θ < π/2. We prove that the resulting discretization is stable, under the assumption ‖(A(t) − A(s))x‖ ≤ L|t− s|α(‖x‖+ ‖A(s)x‖), x ∈ D, t, s ∈ J, where L > 0 and α ∈ (0, 1). Our results are applicable to the analysis of parabolic problems in the Lp, p 6= 2, norms.