Singular limit of a degenerate chemotaxis-Fisher equation
The paper deals with a degenerate parabolic equation that arises in chemotaxis-growth models \[ \frac{\partial u}{\partial t} = \varepsilon \Delta (u^m)- \nabla . (u \nabla v) + \frac{1}{\varepsilon} u(1-u^p) \] in \(\Omega \times (0,T]\) with zero Neumann boundary conditions, initial condition \(u(x,0)=g^\varepsilon (x)\) in \(\Omega \subset \mathbb R^N\), \(m>1\), and a given smooth function \(v(x,t)\). The author studies the behaviour of the (weak) solution to this problem as \(\varepsilon \to 0\). Using the method of sub- and supersolutions, he proves that for suitable initial data \(g^\varepsilon\) with a compact support in \(\Omega\), \(u^\varepsilon \to u^0\) on \([0,T]\) for some \(T>0\). The limiting function \(u^0\) is the characteristic function of a smooth moving domain \(\Omega_t \subset\subset \Omega\), whose motion law is related to \(v\) through the critical velocity of a certain degenerate travelling Fisher wave problem. An example of radially symmetric case with non-trivial solutions is discussed.
scientific article; zbMATH DE number 2111509