Um problema parabólico com condição de fronteira não-linear e peso indefinido: existência, regularidade, bifurcação e estabilidade de equilíbrios
Resumo
This work is concerned with a parabolic problem, occuring in population genetics,
under a nonlinear Neumann boundary condition with a weight of indefinite sign and
a positive parameter. Considering a phase space appropriate to the physical nature
intrinsic to the model, it is proved that the parabolic problem generates a nonlinear
dynamical system, which is a gradient system. Therefore, its equilibrium solutions play
a fundamental role in the long term dynamics. Then the stationary problem is studied
under various aspects: it is proved the existence of a weak equilibrium solution using
the variational method; it is established the regularity of weak equilibrium solutions by
showing that they are classical ones; the bifurcation and stability structures of equilibria
are completely determined. Furthermore the behavior of the trace of the nontrivial
equilibrium solution when the parameter is large is established.