Blow up and growth in the directional solidification of dilute binary alloys

Abstract

Numerous asymptotic equations have been derived to describe the evolution of the solid-liquid interface which occurs during the directional solidification of a dilute binary alloy. In the limit of a small distribution coefficient, many of these equations have been of the form [l-4]

where α and K are positive constants. More recently the equation

has also been derived in this context by a modified asymptotic method,which more accurately preserves some of the original nonlinearities [5]. Both of these equations fall into the class of equations

where f(u) is a positive convex function,where r, K and α are constants and where K and α are positive. For (0.1) and (0.2), we demonstraie that if the solution breaks down in finite time, then blowup of the L^∞norm occurs. Furthermore, if K is sufficiently large, a=0, and if the initial data is sufficiently small, then solutions exist globally and decay to zero exponentially.Estimates on the growth of the H ⁻¹norm are also given. Sufficient conditions for blow up for equations (0.1) and (0.2) are derived for certain boundary conditions. Our results support the numerical analysis and conjectures which appeared in [5].

Keywords:

• Blow up
• fourth order parabolic equations
• directional solidification

AMS:

• 35B35
• 35G30
• 35K35