The Nehari manifold for a degenerate logistic parabolic equation

We will present some recent results on the behavior of solutions to a degenerate logistic equation with a nonlinear term of the form \(b(x)f(u)\), where the weight function \(b\) is assumed to be nonpositive. We exploit variational techniques and comparison principle in order to study the evolutionary dynamics. A crucial role is then played by the Nehari manifold, as we note how it changes as the parameter \(\lambda\) in the equation or the function \(b\) vary, affecting the existence and non-existence of stationary solutions. We describe a detailed picture of the positive dynamics and also address the local behavior of solutions near a nodal equilibrium, which sheds some further light on the study of the evolution of sign-changing solutions.

This is a work in collaboration with Juliana Fernandes (University of Rio de Janeiro, Brasil)