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The general quasi-linear autonomous fourth order diffusion equation ut = -[G(u)uxxx + h(u, ux, uxx)]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. [ABSTRACT FROM AUTHOR]

The article discusses a thesis that applies Vessiot theory and exterior calculus to find local solutions of partial differential equations (PDEs) and examines the classes of PDEs represented by vector fields forming Vessiot distribution. Topics discussed include the development and application of a method to find the largest integrable sub-distributions and solutions of PDEs and results about the Frobenius integrability and solution of evolution equations admitting travelling wave solutions.