Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 60, pp. 1-14

Title: Nehari manifold for degenerate logistic parabolic equations

Authors: Juliana Fernandes (Univ. Federal do Rio de Janeiro, Brazil)
         Liliane Maia (Univ. de Brasilia, Brazil)

Abstract:
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 $\lambda$ 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.
 

Submitted May 13, 2025. Published June 10, 2025.
Math Subject Classifications: 35A01, 35B40, 35K58, 35A15.
Key Words: Logistic equation; variational methods; degenerate problem; parabolic equation.