Asymptotic behavior of solutions of biharmonic Stefan’s problem

Introduction

Free boundary value problems arise in the mathematical description of a variety of physical phenomena. They also appear in the theory minimal surfaces and are known as obstacle problems. They are the boundary value problems where the determination of the boundary itself is a part of the solution to be determined. In free boundary value problems, the variable of interest changes its behavior discontinuously across the boundary. Among several examples of free boundary problems, we list a few. 1. Solidification and crystallization of substance when temperature crosses certain threshold value. 2. Dynamics of contact lines and phase transition.

Internal gravity waves.

In the present article we want to study asymptotic analysis of solutions of free boundary problems for the bi-harmonic Stefan's problem in one space variable. Stefan's problem is one of the most studied free boundary value problems and interest in it stems from the study of how phases evolve during evolution of, say for example, melting of ice. One of the most studied models for fourth order diffusion is thin film equation. For non-negative finite mass on appropriate domain self-similarity solutions of the thin film equation adequately explain finite time blow up of solutions as well as self-similar spreading. The blow-up dynamics of the it is governed by the interactions between non-linear second order and fourth order terms. One of the approaches to deal with the thin film equation is to study monotone dissipation of its energy functional. One of the major sources of difficulties for the free boundary problems is that the energy functional is not non-negative and changes sign on the domain of interest and hence finite time blow up and infinite time spreading in its dynamics. Therefore, it becomes of some interest to study positive solutions of fourth order diffusion equations to illuminate the dynamics and asymptotic analysis of the solutions as time becomes infinite.

The following problem studied by Bertozzi is worthy of mention here is fourth order diffusion equation involving Laplace-Beltrami operator ∆ 𝑆 on the surface 𝑆. 𝑢 𝑡 = -𝛥 𝑆 𝛥 𝑆 𝑢, 𝑢(𝑥, 0) = 𝑢 0 .

Energy functional corresponding to the above diffusion equation is given as follows. We have taken up the study of boundary value problems which involve polyharmonic operators especially because they are in some sense limits of thin film equations without lower order perturbations. The idea is that solutions of thin film equations without lower order perturbations converge to obstacle problem for initial boundary value problem for biharmonic partial differential equations. One of the important topics of research in the analysis of partial differential equations is to study these limit processes in various function spaces. One of the sources of difficulty in this study is understanding the dominance of positivity in the solutions of boundary value problems for poly-harmonic equations.

Main Theorem

We consider the following problem which is followed by main theorem of this article. 𝑢 𝑥𝑥𝑥𝑥 = 𝑢 𝑡 , 0 < 𝑥 < 𝑠(𝑡), 𝑡 > 0, (𝑢 𝑥𝑥 ) 𝑥 (0, 𝑡) = 𝑓(𝑡), 𝑓(𝑡) < 0, 𝑡 > 0, 𝑢(𝑥, 0) = 𝜙(𝑥), 𝜙(𝑥) ≥ 0,0 ≤ 𝑥 ≤ 𝑏, 𝜙(𝑏) = 0, 𝑢 𝑥𝑥 (𝑥, 0) = 𝜓(𝑥), 𝜓′(𝑥) ≥ 0,0 ≤ 𝑥 ≤ 𝑏, 𝑢(𝑠(𝑡), 𝑡) = 0, 𝑡 > 0, (𝑢 𝑥𝑥 ) 𝑥 (𝑠(𝑡), 𝑡) = -𝑑𝑠(𝑡) 𝑑𝑡 , 𝑡 > 0.

( 𝑤 𝑥𝑥𝑥𝑥 = 𝑤 𝑡 , 0 < 𝑥 < 𝑠(𝑡), 𝑡 > 0. 𝑤(𝑥, 0) = 𝛷(𝑥), 0 < 𝑥 < ∞, (𝑤 𝑥𝑥 ) 𝑥 (0, 𝑡) = 𝑓(𝑡) -𝜀𝑡 > 0, 𝜀 > 0.

Where 𝛷(𝑥) is 𝜙(𝑥) extended by 0 beyond the interval 0 ≤ 𝑥 ≤ 𝑏. Solution of the above problem is given by

Where 𝑁 is a Neumann function for the underlying problem which is non-negative (see for example (Sweers 1991)) and therefore we have (𝑤 𝜀 ) 𝑥 ≥ 0 and hence 𝑤 𝜀 (𝑥, 𝑡) ≥ 0 which in particular implies 𝑤 𝜀 (𝑠(𝑡), 𝑡) ≥ 0. Consider the function 𝑤 = 𝑤 𝜀 -𝑢 in the interval 0 ≤ 𝑥 ≤ 𝑠(𝑡) and 0 ≤ 𝑡 ≤ 𝑇 for any 𝑇 > 0. On 𝑥 = 𝑠(𝑡) we have 𝑤 ≥ 0 and when 𝑡 = 0 we have 𝑤 = 0. Thus, we have (𝑤 𝑥𝑥 ) 𝑥 = -𝜀 < 0 and therefore 𝑤 ≥ 0. We get 𝑢(𝑥, 𝑡) ≤ 𝑤(𝑥, 𝑡), ∀0 ≤ 𝑥 ≤ 𝑠(𝑡). We have (𝑡 -𝜏) 1/4 ⅆ𝜏 + 𝐴(𝑡 + 1) -1/4 ) (7) for certain constant 𝐴 which depends on 𝜙. Using equation ( 6) and ( 7) in (5) we get the equations stated in the theorem. ◻

Conclusion

The study of the asymptotic behavior of solutions of evolution equations is important mainly because we are interested in identifying if the solution of evolution equation goes to steady state solution. This steady state solution is a solution of a corresponding boundary value problem for elliptic partial differential equation. Several instances are known where instead of stabilizing into steady state, these solutions blow up in finite time and exhibit spreading of the solution and expanding contact lines or surfaces which are free boundaries. This peculiar behavior is essentially attributed to nonpositivity of solutions even for positive data. Stefan's problem corresponds to Steklov boundary conditions and hence maintain nonnegativity of solutions for nonnegative data. For general boundary conditions we are led to consider obstacle problems which are not well understood for poly-harmonic operators. In this article we have considered Stefan's problem of fourth order diffusion in one space dimension which generalizes to Stefan's problem biharmonic operators in higher space dimensions. Though solutions of biharmonic equations in general change sign for general boundary conditions but they maintain positivity of the solution for positive data for Steklov boundary conditions. The conclusion of the theorem given in this article states that the support of the solution increases as 𝑡 1-𝛿 as solution evolves in time and we see spreading of the solution in the biharmonic case.

References

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