The author studies the initial boundary value problem for a quasilinear degenerate parabolic equation of the form \[ b(u)_ t= u_{xx}+ f(u) \qquad \text{in} \qquad x\in\Omega, \quad t>0, \quad \Omega\subset\mathbb{R}, \] under the initial condition \[ u(x,0)=u_ 0(x) \qquad \text{in} \qquad x\in\Omega, \] together with one of the following three types of boundary conditions: (a) The Dirichlet boundary conditions with \(\Omega=(0,L)\), \(u(0,t)= u(L,t)=0\) in \(t>0\); (b) The Neumann boundary conditions with \(\Omega=(0,L)\), \(u_ x(0,t)= u_ x(L,t)=0\) in \(t>0\); (c) The Cauchy problem, namely, \(\Omega=\mathbb{R}\).
In the case of the Dirichlet or Neumann problem, it is shown that if \(f(u)\) grows more rapidly than \(u\), then the blow-up set of a blow-up solution is finite.
In the case of the Cauchy problem, it is assumed that the initial data \(u_ 0(x)\) has a compact support \([0,L]\). The author is interested in the behavior of the interface near the blow-up time \(t=T\) as well as the shape of the blow-up set.