Explicit solutions of a two-dimensional fourth-order nonlinear diffusion equation
A two-dimensional fourth-order nonlinear diffusion equation of the form \[ \frac{\partial u}{\partial t}+\nabla\left(u\nabla\nabla^2u\right)=0 \] is considered. Polynomial solutions of the form \(u(t,x,y)= c(t)\left(1-x^2/a(t)-y^2/b(t)\right)^2\), where the time evolution \(a(t)\), \(b(t)\) is defined parametrically, are constructed. Elliptic (\(a>b>0\)) and hyperbolic (\(a>0,b<0\)) interfaces are discussed. In the case of parabolic interface solutions have the form \(u(t,x,y)=c(t)\left(x-a(t)-y^2/b(t) \right)^2\). Extensions of the obtained solutions to three and higher dimensions are noted.
scientific article; zbMATH DE number 1993298