Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile

We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct the unique self-similar solution, and show that starting from arbitrary initial data, solution orbits converge to the self-similar solution.


On the content of the paper
The Stefan problem has been extensively studied in the past decades.Despite the number of articles and books published on this topic, [11], [8], [10], [9], there are still open problems left, see for instance [1], [3].One of the questions which requires further attention is the long time behavior of the one-phase Stefan problem, where the heat flux is specified at the fixed boundary, namely the Neumann problem : (i) u t (x, t) − u xx (x, t) = 0, t > 0, 0 < x < s(t), (ii) −u x (0, t) = h √ t + 1 , u(s(t), t) = 0, t > 0, (iii) ṡ(t) = −u x (s(t), t), t > 0, (iv) u(x, 0) = u 0 (x), 0 < x < s(0 We stress that this type of boundary condition is reasonable from the modeling view point.Namely, the decay of data presented in (1-ii) is consistent with the parabolic scaling.We choose, however, to shift the initial time by a positive constant, say 1.In this way we avoid an artificial singularity at the initial time t = 0. Let us note that the existence of a unique smooth solution (u, s) to Problem (1) has already been established.We recall the assumptions of this result in Subsection 1.2.Here u may be called the temperature and s is the position of the interface.
Our approach to study the long time behavior of Problem (1) follows a general heuristics saying that the time asymptotics is determined by the steady states (there is none for (1)) or special solutions such as self-similar solutions or travelling waves.In fact, we show in Corollary 1 that there is exactly one self-similar solution (v, σ), which has the form, v(x, t) = U x √ t+1 and σ(t) = ω √ t + 1 for some constant ω and a profile function U. Our main result states that the self-similar solution is attracting.
Let (u, s) be the corresponding solution of Problem (1).Then, Our method of proof is based upon recent results obtained by [4], [3], who use the comparison principle in an essential way.The argument dwells on the possibility of trapping a given solution to (1) between two solutions with known time asymptotic behavior.In order to make this method work we transform (1) to a problem on a bounded domain with the help of similarity variables.The self-similar solution of (1) corresponds to the steady state solution of the transformed system.Its uniqueness is of crucial importance for the proof.
Let us stress the main difference between [4], [3] and the present article.The authors of [4] and [3] present a quite technical proof to show that the space derivative of the solution uniformly converges to its limit as t → ∞.
Here, we completely avoid such a claim, so that our proof is simpler and more direct, which would make our method easier to adapt to a different setting.
We should point out that there are a number of results dealing with the asymptotic behavior of solutions of Stefan problems, mainly in the case of Dirichlet data on the fixed boundary.However, even for Dirichlet data, there are not so many articles besides [4] and [3] simultaneously addressing the behavior of the temperature profile u and the shape of the interface s.

Existence and uniqueness of the solution
Let us stress that (2) is our standing set of assumptions on the initial conditions.Moreover, the condition u 0 (0) > 0 is necessary to construct proper lower solutions, but is not needed in the Proposition below: Proposition 1. Assume that the initial condition u 0 satisfies (2) and that h > 0.Then, there exists a unique classical solution (u, s) of Problem (1) for all t > 0, in the following sense : We refer to [7,Chapter 8,Theorem 2] for the proof of this proposition.Strictly speaking, the original statement in [7] required u 0 to be of class C 1 ; however, we may relax this assumption in view of [2,Theorem 5.1].
The organization of this paper is as follows.In Subsection 2.1 we discuss the existence and uniqueness of the self-similar solution.Subsection 2.2 is devoted to the study of upper and lower solutions as well as to estimates following from monotonicity.In the last Section, Section 3, we present the proof of the convergence result which is based on the comparison principle.
2 Self-similar, lower and upper solutions

Self-similar solution
We start by re-expressing Problem (1) in terms of the self-similar variables.In other words, we set where η = x √ t + 1 and τ = ln(t + 1), to obtain the problem Let us remark that the existence and uniqueness of the stationary solution of problem (3) were given in [12].
Lemma 1.The associated stationary problem to (3), which is given by admits a unique solution given by the pair (U, ω) such that and ω is the unique positive solution of the equation h = x 2 e x 2 4 .
We immediately conclude from this result that

Lower and upper solutions
Similarly as in [3] we define notions of lower and upper solutions of (3).
Definition 1.We say that a pair of smooth functions The following comparison principle is a fundamental tool in our article.
)) be the extensions by zero of the lower (respectively, upper solutions) of (3) corresponding to the data for every η ≥ 0 and τ ≥ 0.
Proof.The proof is rather similar to those presented by [6, Lemma 2.2 and Remark 2.3] and [5,Lemma 3.5].We omit it here.
In fact, we will construct lower and upper solutions, which are independent of time.For this purpose, we present the perturbed stationary problem whose solution is given by the pair (U λ , b λ ), where U λ (η) = h b λ η e −λs 2 /4 ds, for given λ, and b λ is the unique solution to Remark 1.It is easy to see that for every λ > 0 and η in (0, b λ ), U λ ≥ 0, (U λ ) η < 0 and (U λ ) ηη > 0. In particular, U λ is a linear function for λ = 0, and it is a strictly convex function if λ > 0.
Proof.First we show that if λ > 1, solutions of ( 7) are lower solutions.Indeed for all η ∈ (0, b λ ).Now, from (8) and the inequality e x > x for all x > 0, it thus we can choose λ large enough so that b λ < b 0 .
Next we show that we can choose λ such that U λ ≤ u 0 .On the one hand we have that On the other hand, for every 0 < η < b λ , we deduce from the strict convexity of U λ discussed in Remark 1 that Also we remark that and recall that by the hypothesis (2) u 0 (0) > 0. Thus, if we choose λ large enough so that b λ < u 0 (0) M and U λ (0) < u 0 (0), it follows that We conclude that (U λ , b λ ) is a lower solution according to Definition 1.
is an upper solution.Next, we give an additional condition in order to ensure that which implies a similar property for W .

Convergence
In this section, the pair (W λ , b λ ) is the lower solution for a fixed λ given by ( 13) and (W , b) is the upper solution given in (14).We shall write and to denote solutions of (3).

Lemma 3. Positivity and boundedness
There holds : Proof.Repeatedly apply the comparison principle Theorem 2. Proof.We only prove part a).For the sake of simplicity of notation we suppress the lower bar and we write W . From Theorem 2 we deduce that (17) Now, for a fixed s = σ, we consider the pair (W σ , b σ ) where In particular we have W σ (η) ≥ W λ (η) and b σ ≥ b λ .Then we apply again Theorem 2 to deduce that for every τ ≥ 0 Returning to (17), now consider s = τ + σ for τ ≥ 0. It holds that the pair ) is a solution to problem (3) for the initial conditions (18) for every τ > 0. From the uniqueness of the solution we deduce that for all τ ≥ 0 we have (20) Substituting ( 20) in ( 19) we deduce that which completes the proof of part a).
We remark that if (W , b) (resp.(W , b)) is defined in (15) (resp.( 16)), then the Lemmas 3 and 4 imply that for every λ > 0 In addition, the Lemmas 3 and 4 imply the convergence of W and W , namely and Finally we state and prove the main result of this paper, which in turn implies the result of Theorem 1.
Then, W (•, τ ) converges uniformly to W ∞ , where Proof. .Step 1.We have to identify the limits W ∞ and W ∞ , and improve the convergence.For this purpose, we shall show the estimate, where W = W or W = W are the functions defined in ( 15) and ( 16), respectively.
In principle, we do not know if W ηη and W τ are square integrable over Ω T,1 := {(η, τ ) : η ∈ (0, b(τ )), τ ∈ (T, T + 1)}.This is why we set and ϕ δ ∈ W 1,∞ ((0, ∞)), where Finally, we set φ δ (η, τ ) = ψ δ (η, τ )ϕ δ (τ ).Now, let us multiply the equation (3) − (i) by φ δ W and integrate over Ω T,1 .We arrive at We first analyze the left-hand-side, we see that integration by parts yields, The boundary terms vanish, because the support of φ δ does not intersect ∂Ω T,1 .Now, we want to compute the limit L 1 (T ) = lim δ→0 + L δ 1 (T ).We remark that For the limit in the first integral, note that from the continuity of the integrand, we can apply the mean value property for integrals to obtain that Applying the condition at the moving boundary, we deduce that As for the other terms, we proceed in a similar way to deduce that and that Hence, we conclude that Next we consider the term R δ 1 (T ).We integrate by parts to obtain Again here, the boundary terms vanish, because the support of φ δ is contained in Ω T,1 .We remark that the same argument as above leads us to where we have also used the boundary condition (3−ii) as well as the condition at the interface (3−iii).Similarly one can show that Hence, we conclude In view of Lemma 3 a possible choice of the constant M 1 is given by Step 2. Applying the mean value theorem for integrals in (24) we deduce from Step 1 that there exist two sequences of points τ , it follows that (25) implies the bounds for all β ∈ (0, b ∞ ).Hence, we can select subsequences (not relabelled) such that Since the limit is unique, we deduce that ψ β and Ψ do not depend on the choice of the sequence τ n and using ( 22) and ( 23) we conclude that ψ β is the weak derivative of W ∞ η in every interval (0, β) and that Indeed, in view of Lebesgue monotone convergence theorem we have Step 3. We claim that (W ∞ , b ∞ ) and (W ∞ , b ∞ ) are both stationary solutions, namely solutions of (4).Hence they are smooth and equal.Indeed, we multiply the equation (3-i) by φ ∈ C ∞ c (R) such that φ η (0) = 0 and we integrate on Ω T,1 .Proceeding as in step 2 we set where W = W or W = W .We then deduce from Lebesgue's dominated convergence theorem that lim T →∞ L 2 (T ) = 0. Next, we investigate the right-hand-side R 2 (T ).Integration by parts yields Now, we pass to the limit as T → ∞.It follows from Lebesgue's dominated convergence theorem that lim In addition, Finally, we collect all the results concerning R 2 (T ), while keeping in mind that lim T →∞ L 2 (T ) = 0.This yields for all smooth functions φ in R such that φ η (0) = 0.In particular W ∞ satisfies the differential equation (4-i) in the sense of distributions.
Then, (29) becomes Suppose that φ(b ∞ ) ̸ = 0. Then (30) implies that We deduce that the solution pair (W ∞ , b ∞ ) coincides with the unique solution of Problem (4) or in other words with the unique steady state solution of the time evolution problem, Problem (3).
Step 5. We recall that, in view of step 4,

Figure 1 :
Figure 1: Lower and upper solutions