Finite speed of propagation of perturbations for the Cahn-Hilliard equation with degenerate mobility (Q2764273)

The authors consider two dimensional Cahn-Hilliard model with degenerate mobility. They are interested in solutions with radial symmetric property which satisfy \({\partial (ru) \over \partial t} + {\partial J\over \partial r}=0\) in \((0,R)\times(0,T)\), where \(J= r |u |^n [ k {\partial V \over \partial r} -A'(u) {\partial u\over \partial r}] \) and \(rV = {\partial \over \partial r} (r {\partial u \over \partial r})\) (here \(A(s)\) may be e.g. a cubic like polynomial with a positive top order coefficient). Considering zero boundary conditions for both \(\partial u\over \partial r\) and \(J\) at \(r=0\) and \(r=R\) they show that, for suitably chosen initial condition \(u_0\), \(0<n<1\), and large \(k\), there exists at least one nonnegative weak solution \(u\) such that \(\sup \operatorname {supp} u(\cdot, t) \leq r_1(t)\), \(t\in(0,T)\); \(r_1(t)\) being a continuous increasing function which satisfies \(r_1(0)<R\).