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We consider an abstract time-dependent, linear parabolic problem  \[    u ′ ( t ) = A ( t ) u ( t ) , u ( t 0 ) = u 0 , u’(t) = A(t)u(t), \qquad u(t_0) = u_0,     \]  where      A ( t ) : D ⊂  X →  X A(t) : D \subset X \to X      ,      t ∈  J t \in J      , is a family of sectorial operators in a Banach space      X X       with time-independent domain      D D      . This problem is discretized in time by means of an A(     θ  \theta      ) strongly stable Runge-Kutta method,      0 &gt; θ  &gt; π  / 2 0 &gt; \theta &gt;\pi /2      . We prove that the resulting discretization is stable, under the assumption  \[    ‖  ( A ( t ) −  A ( s ) ) x ‖  ≤  L | t −  s | α  ( ‖  x ‖  + ‖  A ( s ) x ‖  ) , x ∈  D , t , s ∈  J , \| (A(t) - A(s) )x \| \le L|t-s|^\alpha (\|x\|+ \| A(s)x\|), \qquad x\in D, \,t,\,s \in J,     \]  where      L &gt; 0 L&gt;0       and      α  ∈  ( 0 , 1 ) \alpha \in (0,1)      . Our results are applicable to the analysis of parabolic problems in the      L p L^p      ,      p ≠  2 p \ne 2      , norms.

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