On stability of two degenerate reaction–diffusion systems

In this paper, we construct a partially degenerate reaction–diffusion equation subject to the Neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable. In this way, we solve an open problem proposed by Casten and Holland (1977) [4]. Moreover, we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross-diffusion effects, which cannot be determined by a simple spectral analysis based on the well developed semigroup theory.