Ill-posedness issue for a multidimensional hyperbolic-parabolic model of chemotaxis in critical Besov spaces B˙2d,1−32×(B˙2d,1−12)d

Abstract In this paper, by fully exploiting nonlinear structure of the cross term in the chemotaxis system and using microlocal analysis, we prove that this system is ill-posed for the critical case B ˙ 2 d , 1 − 3 2 × ( B ˙ 2 d , 1 − 1 2 ) d in the sense of “norm inflation”, which is an unsolved problem in [21] . This result improves the ill-posedness theories of this system in critical Besov spaces framework.