This paper analyses front propagation of the equationwhere f is a monostable (i.e. Fisher-type) nonlinear reaction term and D(v) changes its sign once, from positive to negative values, in the interval v ∈ [0, 1] where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density v at time τ and position x. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states v ≡ 0 and v ≡ 1. In the degenerate case, i.e. when D(0) = 0 and/or D(1) = 0, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.

Front propagation for the aggregation-diffusion-reaction equation [Formula: see text] is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation.

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We study the existence of positive solutions on the half-line [0, ∞) for the nonlinear second order differential equationsatisfying Dirichlet type conditions, say x(0) = 0, lim t→∞ x(t) = 0. The function b is allowed to change sign and the nonlinearity F is assumed to be asymptotically linear in a neighborhood of zero and infinity. Our results cover also the cases in which b is a periodic function for large t or it is unbounded from below.Keywords. Second order nonlinear differential equation, boundary value problem on the half line, Dirichlet conditions, globally positive solution, disconjugacy, principal solution.MSC 2010: Primary 34B40, Secondary 34B18.