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This work concerns with a class of chemotaxis models in which external sources, comprising nonlocal and gradient-dependent damping reactions, influence the motion of a cell density attracted by a chemical signal. The mechanism of the two densities is studied in bounded and impenetrable regions. In particular, it is seen that no gathering effect for the cells can appear in time provided that the damping impacts are sufficiently strong.

This paper investigates the properties of classical solutions to a class of chemotaxis systems that model interactions between tumor and immune cells. Our focus is on examining the global existence and explosion of such solutions in bounded domains of $\mathbb{R}^n$, $n\geq3$, under Neumann boundary conditions. We distinguish between two scenarios: one where all equations are parabolic and another where only one equation is parabolic while the rest are elliptic. Boundedness is demonstrated under smallness assumptions on the initial data in the former scenario, while no such constraints are necessary in the latter. Additionally, we provide estimates for the blow-up time of unbounded solutions in three dimensions, supported by numerical simulations.

This article deals with a class of chemotaxis systems describing mechanisms from mathematical biology. In the specific, for a rather general class of attraction-repulsion models, with nonlinear productions, diffusion, sensitivities and logistic term, we are interested in deriving interplays on the parameters involved in the problem capable to ensure globality and boundedness of related solutions. This project is precisely contextualized in the frame of a series of results dealing with local and nonlocal, and linear and nonlinear, attraction-repulsion chemotaxis systems. For these problems, rooms of improvements are still open; in this sense, this research exactly enhances and extends already known analyses in the literature. More specifically, a direct comparison with [Jiao, Jadlovsk\'a, Li, Nonlinear Anal. Real World Appl., 2023; Ren, Liu, Math. Models Methods Appl. Sci., 2020; Chiyo, Yokota, Z. Angew. Math. Phys., 2022; Columbu, Frassu, Viglialoro, Appl. Anal., 2023] is carried out.

In this paper we propose a method for computing the contour of an object in an image using a snake represented as a subdivision curve. The evolution of the snake is driven by its control points which are computed minimizing an energy that pushes the snake towards the boundary of the interest region. Our method profits from the hierarchical nature of subdivision curves, since the unknowns of the optimization process are the few control points of the subdivision curve in the coarse representation and, at the same time, good approximations of the energies and their derivatives are obtained from the fine representation. We introduce a new region energy that guides the snake maximizing the contrast between the average intensity of the image within the snake and over the complement of the snake in a bounding box that does not change during the optimization. To illustrate the performance of our method we discuss the snakes associated with two classical subdivision schemes: the four point scheme and the cubic B-spline. Our experiments using synthetic and real images confirm that the proposed method is fast and robust.