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A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. (English) Zbl 1360.35092

Summary: The Neumann boundary value problem for the chemotaxis system generalizing the prototype \[ \begin{cases} u_t=\nabla\cdot( D(u)\nabla u)-\nabla\cdot(u\nabla v), x\in \Omega, t>0,\\ v_t=\Delta v-uv, x\in \Omega, t>0, \end{cases}\tag{KS} \] is considered in a smooth bounded convex domain \(\Omega\subset \mathbb{R}^N(N\geq2)\), where \[ D(u)\geq C_D(u+1)^{m-1}\,\,\text{for all}\,\,u\geq 0\,\,\text{with some}\,\,m>1\,\,\text{and}\,\,C_D>0. \] If \( m>\frac{3N}{2N+2}\) and suitable regularity assumptions on the initial data are given, the corresponding initial-boundary problem possesses a global classical solution. Our paper extends the results of L. Wang et al. [Z. Angew. Math. Phys. 66, No. 4, 1633–1648 (2015; Zbl 1328.92019)], who showed the global existence of solutions in the cases \(m>2-\frac{6}{N+4}\) (\(N\geq3\)). If the flow of fluid is ignored, our result is consistent with and improves the result of Y. Tao and M. Winkler [Discrete Contin. Dyn. Syst. 32, No. 5, 1901–1914 (2012; Zbl 1276.35105); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30, No. 1, 157–178 (2013; Zbl 1283.35154)], who proved the possibility of global boundedness, in the case that \(N=2,m>1\) and \(N= 3\), \(m > \frac{8}{7}\), respectively.

MSC:

35K55 Nonlinear parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35Q35 PDEs in connection with fluid mechanics
92C17 Cell movement (chemotaxis, etc.)
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