| Abstract: |
| We analyse 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. |
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