The Nehari manifold for a degenerate logistic parabolic equation

Abstract

The present paper analyses 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 λ 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.