Abstract
We consider a quasilinear chemotaxis–consumption system with singular sensitivity \(u_t=(D(u)u_x)_x-\chi (\frac{S(u)}{v} v_x)_x\), \(v_t=v_{xx}-uv\) in a bounded interval \(\Omega \subset {\mathbb {R}}\) with \(\chi >0\). The diffusivity fulfills \(D(u)\ge D_0(u+1)^{m-1}\) with \(D_0,m>0\), and the density-signal governed sensitivity satisfies \(0<S(u)\le D_1(u+1)^M\) with \(D_1, M>0\). It is proved that the system possesses a globally bounded classical solution under one dimension if \(M<\frac{2}{3}+\frac{5m}{6}\) for \(0<m \le 1\) or \(M<1+\frac{m}{2}\) for \(m>1\).
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Acknowledgements
This work was supposed by the Doctoral Scientific Research Foundation of Liaoning Normal University (Grant No. 203070091907), the Doctoral Scientific Research Foundation of Liaoning Science and Technology Department (Grant No. 2020-BS-185).
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Zhao, X. Boundedness to a quasilinear chemotaxis–consumption system with singular sensitivity in dimension one. Z. Angew. Math. Phys. 72, 185 (2021). https://doi.org/10.1007/s00033-021-01614-7
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DOI: https://doi.org/10.1007/s00033-021-01614-7