Stability of Runge-Kutta methods for quasilinear parabolic problems

We consider a quasilinear parabolic problem u′(t) = Q ( u(t) ) u(t), u(t0) = u0 ∈ D, where Q(w) : D ⊂ X → X, w ∈ W ⊂ X, is a family of sectorial operators in a Banach space X with fixed domain D. This problem is discretized in time by means of a strongly A(θ)-stable, 0 < θ ≤ π/2, Runge–Kutta method. We prove that the resulting discretization is stable, under some natural assumptions on the dependence of Q(w) with respect to w. Our results are useful for studying in Lp norms, 1 ≤ p ≤ +∞, many problems arising in applications. Some auxiliary results for time-dependent parabolic problems are also provided.