Blow-up and global existence for a kinetic equation of swarm formation

Abstract

In the present paper we study possible blow-ups and global existence for a kinetic equation that describes swarm formations in the variable interacting rate case.

Modeling Asymmetric Interactions in Economy

“…The mathematical theory of such structures is much more complex than the corresponding theory for equations with symmetric kernels-cf. [6,18,[22][23][24]. In fact, at the formal level, we showed the possible singular behavior of the solutions.…”

“…In contrast to the previous attempts [6,18,[22][23][24], in the present paper, we study an asymmetric way of interactions. The symmetric interactions resulting in nonlinear diffusion equations were considered in reference [24].…”

“…We note that similar general structures were proposed by Othmer et al in [19] and studied, for example, in [20,21] (see also references therein). A detailed bibliographical review can be found in [22]. Various kinetic equations were discussed in [1].…”

“…The simplest choice could be γ = 1; however, the case γ > 1 leads to more difficult considerations involving possibilities of blow-ups in a finite time, see [22][23][24]. The previous studies of Equation (4) were concentrated on symmetric interactions, that is the symmetric function κ.…”

Mathematical Analysis of an Autoimmune Diseases Model: Kinetic Approach

“…The KTAP has been actively used for modeling phenomena not only in physics (see e.g., Reference [39,40]) but also in many other areas of applied and life sciences, for example in the political [41], economic and social [42][43][44][45][46][47] systems, psychological interactions [48], the dynamics of crowds [49][50][51][52][53][54] and swarms [55][56][57][58][59][60][61], traffic flow [62][63][64][65][66][67][68] and so forth.…”