Degenerate diffusion equations with drift functionals modelling aggregation. (English) Zbl 0594.35061
The author considers 1-dimensional diffusive random walks with density dependent diffusion coefficient \(\mu =\mu (u)\geq 0\) and a drift \(\gamma =\gamma (t,x,u)\) described by \(\partial_ tu=\partial_ x\{\mu (u)\partial_ xu-\gamma (\cdot,u)\}\) for the density u(t,x), \(t\geq 0\), \(t\in I\). The interesting point of this paper is that he studies the case, that the diffusion coefficient \(\mu =\mu (u)\) degenerates not for low density \(u=0\), but for some saturating density \(u=a>0\) where, for example, \(\mu (u)>0\) \((0\leq u<a)\), \(\mu (a)=0\), \(-\infty <\mu '(a)<0\).
Reviewer: Y.Ebihara
MSC:
| 35K65 | Degenerate parabolic equations |
| 92D25 | Population dynamics (general) |
| 65N99 | Numerical methods for partial differential equations, boundary value problems |
Keywords:
degenerate diffusion; aggregation models; Lyapunov function; 1- dimensional diffusive random walks; density dependent diffusion coefficientReferences:
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