Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation
The authors study nonnegative solutions of the equation \(u_t + (u^nu_{xxx})_x = 0\), where \(n\) is a real positive constant; the equation is degenerated at points \(x\) at which \(u\) vanishes. For \(0 < n < 1/2\), it is shown that initially strictly positive solutions may vanish at some point \(x_0\) after a finite time \(t_0\). This result is of particular interest for several hydrodynamic applications in which the above equation arises. For example, if \(u\) is interpreted as the thickness of a liquid film, it implies that at time \(t_0\) the film breaks at \(x_0\).
scientific article; zbMATH DE number 760206