2024/09/03 by Zeng, Ziyue, Li, Yuxiang
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.01853
This paper investigates the repulsion-consumption system \ ut=Δu+∇ ⋅(S(u) ∇ v), τvt=Δv-u v, . under no-flux/Dirichlet conditions for u and v in a ball BR(0) ⊂ \mathbb Rn . When τ=\0,1\ and 00, we show that for any given radially symmetric initial data, the problem (⋆) possesses a global bounded classical solution. Conversely, when τ=0, n=2 and S(u) \geqslant k uβ for u \geqslant 0 with some β>1 and k>0, for any given initial data u0, there exists a constant M⋆=M⋆(u0)>0 with the property that whenever the boundary signal level M\geqslant M⋆, the corresponding radially symmetric solution blows up in finite time. Our results can be compared with that of the papers [J.~Ahn and M.~Winkler, \it Calc. Var. \bf 64 (2023).] and [Y. Wang and M. Winkler, \it Proc. Roy. Soc. Edinburgh Sect. A, 153 (2023).], in which the authors studied the system (⋆) with the first equation replaced respectively by ut=∇ ⋅ ((1+u)-α ∇ u)+∇ ⋅(u ∇ v) and ut=∇ ⋅ ((1+u)-α ∇ u)+∇ ⋅((u)/(v) ∇ v). Among other things, they obtained that, under some conditions on u0(x) and the boundary signal level, there exists a classical solution blowing up in finite time whenever α>0.