Asymptotic behavior of solutions of biharmonic Stefan’s problem

Abstract

In this article we have considered Stefan’s problem in one dimensional situation for biharmonic operator. Stefan’s problem is very important from the point of view of the study of free boundary value problems and spreading of solutions. Blow up in finite time and spreading of solutions are still less understood phenomena and it becomes necessary to look for and analyze models which offer similar behavior but are significantly different from the traditional Stefan’s problem. The elliptic part of Stefan’s problem for biharmonic case corresponds to Steklov boundary value problem. In general, Green’s function and Neuman’s functions change sign on domain of interest. But, for Steklov boundary conditions these fundamental solutions do not change sign and this property is used to prove asymptotic behavior of the free boundary for the biharmonic free boundary. We have proved that free boundary for Stefan’s problem corresponding to biharmonic operator grows like t^(1-δ) for δ=1/4.