Fourth Order Diffusion Equations with Increasing Entropy

The general quasi-linear autonomous fourth order diffusion equation u(t) = -[G(u)u(xxx) + h(u, u(x), u(xx))](x) with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may have any positive twice-differentiable diffusivity function G(u). The forms of such equations are the indicators of more general conservation equations whose entropy equation may be expressed in an alternative reaction-diffusion form whose source term, although reducible, is positive.