High-Order Nonlinear Diffusion

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Abstract
A number of important physical processes, such as the flow of a surface-tension dominated thin liquid and the diffusion of dopant in semiconductors are governed by the fourth-order nonlinear diffusion equation ut + ( unuxxx ) x =0 ( n >0). The analysis of such equations is doubly complex due to the nonlinearity and the high order. Here we present some of the more immediate and simple results for this equation and our development parallels known results for the classical nonlinear diffusion equation ut = ( unux ) x . In particular we examine in some detail asymptotic solutions for small n (namely 0< n ≪1), simple waiting-time solutions and the similarity source solution. The similarity source solution for n =1 has a particularly simple closed form which may be readily generalized to a nonlinear diffusion equation of arbitrary high order. This simple exact similarity solution appears not to have been noted previously and together with the well-known solution for the classical equation means that the general nonlinear equation ut =(−1) m D( u n D 2 m +1 u ), with D=∂/∂ x , admits a simple exact similarity solution either for m =0 and all values of n or for n =1 and all values of m . Details for small- n solutions and waiting-time solutions for this general equation are also briefly noted.

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