论文标题
定量强的抛物线最大原理，并应用于出租车型迁移消耗模型，涉及信号依赖性退化扩散
A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion
论文作者
论文摘要
\ [ u_t =δ\ big（ux（v）\ big）， v_t =ΔV-uv， \ qquad（*）\是为了适当平滑功能$ ϕ：[0，\ infty）\ to r $，即$（0，\ infty）$上的$ ϕ> 0 $，但此外，$ ϕ（0）= 0 $ with $ ding $ with $ ϕ'（0）> 0 $。 为了适当应付包括的扩散变性，本研究分别研究了线性方程的诺伊曼问题\ [ v_t =ΔV + \ nabla \ cdot \ big（a（x，t）v \ big） + b（x，x，t）v \]，并建立了一个关于非负溶液的点置正界的陈述，取决于超级和$ a $和$ b $的初始数据和$ a $ a $ a $和$ b $的质量。 此后，在仅在两个组件中合适的正常初始数据均不成立的情况下，这是在（*）中衍生出（*）的全局解决方案的结果，平滑而经典的结果的关键工具。除此之外，这些溶液还可以稳定在某些平衡方面，并且由于扩散的变性，作为定性效应，第二个成分的初始较小性的标准被确定为该极限状态以使其在空间上是非构造的。
The taxis-type migration-consumption model accounting for signal-dependent motilities, as given by \[ u_t = Δ\big(uϕ(v)\big), v_t = Δv-uv, \qquad (*) \] is considered for suitably smooth functions $ϕ:[0,\infty)\to R$ which are such that $ϕ>0$ on $(0,\infty)$, but that in addition $ϕ(0)=0$ with $ϕ'(0)>0$. In order to appropriately cope with the diffusion degeneracies thereby included, this study separately examines the Neumann problem for the linear equation \[ V_t = ΔV + \nabla\cdot \big( a(x,t)V\big) + b(x,t)V \] and establishes a statement on how pointwise positive lower bounds for nonnegative solutions depend on the supremum and the mass of the initial data, and on integrability features of $a$ and $b$. This is thereafter used as a key tool in the derivation of a result on global existence of solutions to (*), smooth and classical for positive times, under the mere assumption that the suitably regular initial data be nonnegative in both components. Apart from that, these solutions are seen to stabilize toward some equilibrium, and as a qualitative effect genuinely due to degeneracy in diffusion, a criterion on initial smallness of the second component is identified as sufficient for this limit state to be spatially nonconstant.
