THE LARGE-TIME SOLUTION OF A NONLINEAR FOURTH-ORDER EQUATION INITIAL-VALUE PROBLEM I. INITIAL DATA WITH A DISCONTINUOUS EXPANSIVE STEP

Publisher： Cambridge University Press

E-ISSN： 1446-8735|51|2|178-190

ISSN： 1446-1811

Source： ANZIAM Journal, Vol.51, Iss.2, 2009-10, pp. : 178-190

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Abstract

In this paper we consider an initial-value problem for the nonlinear fourth-order partial differential equation u_t+uu_x+γu_xxxx=0, −∞<x<∞, t>0, where x and t represent dimensionless distance and time respectively and γ is a negative constant. In particular, we consider the case when the initial data has a discontinuous expansive step so that u(x,0)=u₀(>0) for x≥0 and u(x,0)=0 for x<0. The method of matched asymptotic expansions is used to obtain the large-time asymptotic structure of the solution to this problem which exhibits the formation of an expansion wave. Whilst most physical applications of this type of equation have γ>0, our calculations show how it is possible to infer the large-time structure of a whole family of solutions for a range of related equations.