  Explicit Runge-Kutta methods are classical and widespread techniques in the
numerical solution of ordinary differential equations (ODEs). Considering
partial differential equations, spatial semidiscretisations can be used to
obtain systems of ODEs that are solved subsequently, resulting in fully
discrete schemes. However, certain stability investigations of high-order
methods for hyperbolic conservation laws are often conducted only for the
semidiscrete versions. Here, strong stability (also known as monotonicity) of
explicit Runge-Kutta methods for ODEs with nonlinear and semibounded (also
known as dissipative) operators is investigated. Contrary to the linear case,
it is proven that many strong stability preserving (SSP) schemes of order two
or greater are not strongly stable for general smooth and semibounded nonlinear
operators. Additionally, it is shown that there are first order accurate
explicit SSP Runge-Kutta methods that are strongly stable (monotone) for
semibounded (dissipative) and Lipschitz continuous operators.
