Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system

“…Recently, for a bounded convex domain Ω ⊂ R n , Wang et al [29,30] investigated the global existence and boundedness of solutions to system (1.1) with nonlinear diffusion. Precisely, when D(u) ≥ D 0 (u + 1) m−1 with some constant D 0 > 0 and m > 1 2 (for n = 1) and m > 2 − 2 n (for n ≥ 2), Wang et al [29] proved the system (1.1) admits a unique global bounded classical solution.…”

“…Precisely, when D(u) ≥ D 0 (u + 1) m−1 with some constant D 0 > 0 and m > 1 2 (for n = 1) and m > 2 − 2 n (for n ≥ 2), Wang et al [29] proved the system (1.1) admits a unique global bounded classical solution. For n ≥ 3, Wang et al [30] further proved that system (1.1) with D(u) ≥ D 0 u m−1 and m > 2 − 6 n+4 has a unique global classical solution for non-degenerate diffusion case while it admits at least one global weak solution for degenerate diffusion case. Notice that n ≥ 3 implies that 2− 6 n+4 < 2− 2 n and thus [30] extends the range of nonlinear diffusion obtained in [29] in respect of the global existence of solution.…”

“…For n ≥ 3, Wang et al [30] further proved that system (1.1) with D(u) ≥ D 0 u m−1 and m > 2 − 6 n+4 has a unique global classical solution for non-degenerate diffusion case while it admits at least one global weak solution for degenerate diffusion case. Notice that n ≥ 3 implies that 2− 6 n+4 < 2− 2 n and thus [30] extends the range of nonlinear diffusion obtained in [29] in respect of the global existence of solution. However, this remains a gap for the uniform boundedness of solutions.…”

“…[30] in the sense that Theorem 1.1 establishes the global boundedness of weak solutions. We also removed the convexity assumption on the domain used by [29,30].…”

Global boundedness in a quasilinear chemotaxis model with nonlinear diffusion and consumption of chemoattractant

“…In Wang et al, the authors proposed that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in a convex smooth bounded domain , provided that some technical conditions are fulfilled. When χ ( v ) = 1 and g ( v ) = v , Wang et al improved a previous result in the previous study on global existence of solutions in a convex smooth‐bounded domain, , under the condition that can be relaxed to . In Wang and Xiang, the authors obtained that the system possesses global bounded weak solutions for any sufficiently smooth nonnegative initial data in a bounded domain, , when .…”