Properties of given and detected unbounded solutions to a class of chemotaxis models
Alessandro Columbu, Silvia Frassu, Giuseppe Viglialoro
Abstract
Abstract This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow‐up time are established. Finally, for a simplified version of the model, some blow‐up criteria are proved. More precisely, we analyze a zero‐flux chemotaxis system essentially described as The problem is formulated in a bounded and smooth domain Ω of , with , for some , , , and with . A sufficiently regular initial data is also fixed. Under specific relations involving the above parameters, one of these always requiring some largeness conditions on , we prove that any given solution to (), blowing up at some finite time becomes also unbounded in ‐norm, for all ; we give lower bounds T (depending on ) of for the aforementioned solutions in some ‐norm, being ; whenever , we establish sufficient conditions on the parameters ensuring that for some u 0 solutions to () effectively are unbounded at some finite time. Within the context of blow‐up phenomena connected to problem (), this research partially improves the analysis in Wang et al. ( J Math Anal Appl . 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic.