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We investigate an unstirred chemostat model in which two species compete in a two-dimensional environment. The populations are assumed to disperse anisotropically, with distinct probabilities assigned to horizontal and vertical movements, which are interpreted as dispersal strategies. First, we analyze the dynamics of the single-species model and identify the conditions for the existence of positive steady states. Then, we classify the dynamical behavior of the two-species model into three scenarios based on the diffusion strategies: (i) extinction; (ii) competitive exclusion; and (iii) coexistence. Next, we provide sufficient conditions for the existence of coexistence steady states. Finally, our numerical simulations provide visual validation of our theoretical results and offer valuable insights for future researches.

Within the generalized Newton-Cartan theory, Galilean Twisted spacetimes are introduced as dual models of the well-known relativistic twisted spacetimes. As a natural generalization, torqued vector fields in Galilean spacetimes are defined, showing that the local structure of a Galilean spacetime admitting a timelike torqued vector field is given by a Twisted spacetime. In addition, several results assuring the global splitting as Twisted spacetime are obtained. On the other hand, completeness of free falling observers is studied, as well as geodesic completeness.

In this paper, we are interested in the existence of a positive solution of the two coupled system of nonlinear Schrödinger and Choquard equations. Our equations admit the case that the nonlinearity exponents of two components are different. This includes, in particular, the upper and lower critical cases. We first investigate the existence of ground states and provide a criterion for two-component positiveness of ground states. In case where the ground state is semi-trivial, we construct a higher energy positive solution of the mountain-pass type on the Nehari manifold.

In this paper, we study the existence and nonexistence results for a class of pseudo-parabolic equations with combined logarithmic nonlinearities γ | u | p- 2 u  log   | u | + u  log  | u |, where γ > 0 and p > 2 satisfy certain conditions. Employing the key ingredients in modified potential well method, we obtain the existence of a global weak solution and the finite time blow-up phenomenon for low initial energy and critical initial energy. We show that the global weak solution has exponential decay. An interesting observation is that the higher power term γ | u   | p- 2 u  log  | u | ( p > 2) breaks the infinite time blow-up phenomenon of u  log  | u | with appropriate coefficient γ . Furthermore, an upper bound and a lower bound for the blow-up time are estimated. We give sufficient conditions for global existence and blow-up result of weak solutions in the case of high initial energy. Then we take account of the case when the initial energy is independent of the well depth. In addition, we show that the solution to the same pseudo-parabolic equation with only logarithmic nonlinearity u  log  | u | blows up at infinity.