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This paper proves the uniqueness of the positive linearly stable steady-state for a paradigmatic class of superlinear indefinite parabolic problems arising in population dynamics, under non-homogeneous Dirichlet conditions on the boundary of the domain. The result is absolutely non-trivial, since examples are known for which the model admits an arbitrarily large number of steady-states. Our proof is based on some local and global continuation techniques. Optimal existence and multiplicity results are also obtained through some additional monotonicity and topological techniques.
uniqueness of linearly stable steady-states; super linear indefinite problems; a priori bounds; optimal multiplicity