Properties of given and detected unbounded solutions to a class of chemotaxis models

Columbu, A;Frassu, S

;Viglialoro, G

2023-01-01

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Abstract

his 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 ⋄ (Formula presented.) The problem is formulated in a bounded and smooth domain Ω of (Formula presented.), with (Formula presented.), for some (Formula presented.), (Formula presented.), (Formula presented.), and with (Formula presented.). A sufficiently regular initial data (Formula presented.) is also fixed. Under specific relations involving the above parameters, one of these always requiring some largeness conditions on (Formula presented.), (i)we prove that any given solution to ((Formula presented.)), blowing up at some finite time (Formula presented.) becomes also unbounded in (Formula presented.) -norm, for all (Formula presented.); (ii)we give lower bounds T (depending on (Formula presented.)) of (Formula presented.) for the aforementioned solutions in some (Formula presented.) -norm, being (Formula presented.); (iii)whenever (Formula presented.), we establish sufficient conditions on the parameters ensuring that for some u0 solutions to ((Formula presented.)) effectively are unbounded at some finite time. Within the context of blow-up phenomena connected to problem ((Formula presented.)), 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.