Nehari manifold for degenerate logistic parabolic equations - Juliana Fernandes; Liliane Maia (Electronic Journal of Differential Equations)
In this article we analyze the behavior of solutions to a degenerate logistic equation with a nonlinear term $b(x)f(u)$ where the weight function $b$ is non-positive. We use variational techniques and the comparison principle 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.