Electron. J. Differential Equations, Vol. 2025 (2025), No. 60, pp. 1-14

Nehari manifold for degenerate logistic parabolic equations

Juliana Fernandes, Liliane Maia

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.
DOI: 10.58997/ejde.2025.60

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Juliana Fernandes
Universidade Federal do Rio de Janeiro
Instituto de Matemática
21941-909 Rio de Janeiro - RJ, Brazil
email: jfernandes@im.ufrj.br
Liliane de A. Maia
Universidade de Brasilia
Departamento de Matem\ática
70.910-900, Brasíllia, DF, Brazil
email: lilimaia@unb.br

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