Multidimensional exact solutions to a quasilinear parabolic equation with anisotropic heat conductivity.
Multiple existence and stability of steady-states for a prey-predator system with cross-diffusion
This article discusses a prey-predator system with cross-diffusion. We obtain multiple positive steady-state solutions of this system. More precisely, we prove that the set of positive steady-states possibly contains an S or ⊃-shaped branch with respect to a bifurcation parameter in the large cross-diffusion case. Next we give some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case. Our...