Abstract
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.
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1 Introduction
1.1 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, [8,9,10,11], 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:
where we assume that \(h>0\).
We stress that this type of boundary condition is reasonable from the modeling view point. Namely, the choice of h means that the water in the container is heated at \(x=0\) that leads to ice melting at \(x= s(t)\). We choose (1\(-ii\)), because it simple, yet it leads to non-trivial behavior of solutions.
The decay of data presented in (1\(-ii\)) is consistent with the parabolic scaling, i.e. when we change variables by introducing \(\eta =\frac{x}{\sqrt{t+1}}\), then the first condition in (1\(-ii\)) will be transformed to a constant in time, see the first condition in (3\(-ii\)). Note that we have shifted the initial time by 1, in order to 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 \((v,\sigma )\), which has the form, \(v(x,t) = U\left( \frac{x}{\sqrt{t+1}} \right) \) and \(\sigma (t) = \omega \sqrt{t+1}\) for some constant \(\omega \) and a profile function U. Our main result states that the self-similar solution is attracting.
Theorem 1
Suppose that \((u_0, b_0)\) satisfies the conditions
Let (u, s) be the corresponding solution of Problem (1). Then,
-
1)
\(\lim \limits _{t\rightarrow \infty }s(t)/\sqrt{t+1} = \omega \);
-
2)
\(\displaystyle {\lim \limits _{t\rightarrow \infty }\sup _{x/\sqrt{t+1}\in [0,\omega ]} \left| u(x,t) - U\left( \frac{x}{\sqrt{t+1}}\right) \right| } =0.\)
Our method of proof is based upon recent results obtained by [3, 4], 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 [3, 4] and the present article. The authors of [3, 4] present a quite technical proof to show that the space derivative of the solution uniformly converges to its limit as \(t\rightarrow \infty .\) 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 [3, 4] simultaneously addressing the behavior of the temperature profile u and the shape of the interface s.
1.2 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:
When we want to emphasize the dependence of (u, s) on the initial conditions we will write \(u=u(\cdot , \cdot , (u_0, s_0))\), \(s= s(\cdot , (u_0, s_0))\).
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 Sect. 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
2.1 Self-similar solution
We start by re-expressing Problem (1) in terms of the self-similar variables. In other words, we set
where \(\displaystyle {\eta =\frac{x}{\sqrt{t+1}}}\) and \(\tau =\ln (t+1)\), 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 \((U,\omega )\) such that
and \(\omega \) is the unique positive solution of the equation \(h= \displaystyle {\frac{x}{2} e^{\frac{x^2}{4}}}.\)
We immediately conclude from this result that
Corollary 1
If we set \(u(x,t) = U\left( \displaystyle {\frac{x}{\sqrt{t+1}}}\right) \) and \(\sigma = \omega \sqrt{t+1}\), then \((u,\sigma )\) is a solution to (1\(-i\))–(1\(-iii\)).
2.2 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 \((\underline{W},\underline{b})\) (resp. \((\overline{W},\overline{b}))\) is a lower (resp. upper) solution of Problem (3) if
The following comparison principle is a fundamental tool in our article.
Theorem 2
Let \((\underline{W}_1(\eta ,\tau ),\underline{b}_1(\tau ))\) (respectively, \((\overline{W}_2(\eta ,\tau ),\overline{b}_2(\tau ))\)) be the extensions by zero of the lower (respectively, upper solutions) of (3) corresponding to the data \((h_1,{u_0}_1,{b_0}_1)\) (respectively, \((h_2,{u_0}_2,{b_0}_2)\)). If \(h_1 \le h_2\), \({u_0}_1\le {u_0}_2\) and \({b_0}_1\le {b_0}_2\), then \(\underline{b}_1(\tau )\le \overline{b}_2(\tau )\) for every \(\tau >0\) and \(\underline{W}_1(\eta ,\tau )\le \overline{W}_2(\eta ,\tau )\) for every \(\eta \ge 0\) and \(\tau \ge 0\).
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. \(\square \)
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 \((U_\lambda ,b_\lambda )\), where \(U_\lambda (\eta )= {\tilde{h}} \int _\eta ^{b_\lambda } e^{-\lambda s^2/4}\textrm{d}s, \) for given \(\lambda \), and \(b_\lambda \) is the unique solution to
Remark 1
It is easy to see that for every \(\lambda >0\) and \(\eta \) in \((0,b_\lambda )\), \(U_\lambda \ge 0\), \( (U_\lambda )_\eta < 0\) and \((U_\lambda )_{\eta \eta }> 0\). In particular, \(U_\lambda \) is a linear function for \(\lambda =0\), and it is a strictly convex function if \(\lambda >0\).
Lemma 2
Let \((U_\lambda ,b_\lambda )\) be a solution of (7). If \({\tilde{h}} \le h\), then for all \(\lambda > 1\) \((U_\lambda ,b_\lambda )\) is an independent of time lower solution to (3).
Proof
First we show that if \(\lambda > 1\) is sufficiently large, then solutions of (7) are lower solutions. Indeed
for all \(\eta \in (0, b_\lambda )\). Now, from (8) and the inequality \(e^x > x\) for all \(x>0\), it holds that \(0< b_\lambda =2 {\tilde{h}} e^{-\lambda b^2_\lambda /4} \le \displaystyle {\frac{8 {\tilde{h}}}{\lambda b^2_\lambda }}\), which implies that \(b_\lambda \rightarrow 0\) as \(\lambda \rightarrow \infty \); thus we can choose \(\lambda \) large enough so that \(b_\lambda <b_0\).
Next we show that we can choose \(\lambda \) such that \(U_\lambda \le u_0\). On the one hand we have that
On the other hand, for every \(0<\eta <b_\lambda \), we deduce from the strict convexity of \(U_\lambda \) discussed in Remark 1 that
Also we remark that
and recall that by the hypothesis (2) \(u_0(0) > 0\). Thus, if we choose \(\lambda \) large enough so that \(b_\lambda <\displaystyle {\frac{u_0(0)}{M}}\) and \(U_\lambda (0)<u_0(0),\) it follows that
We conclude that \((U_\lambda ,b_\lambda )\) is a lower solution according to Definition 1. \(\square \)
Now, we define the pair \((\underline{W}_\lambda ,\underline{b}_\lambda )\) by
Next we propose an upper solution which is a straight line on its support. It is easy to verify that the pair \((\overline{W},\overline{b})\) defined by
is an upper solution. Next, we give an additional condition in order to ensure that \({\overline{W}} \ge u_0\) on the interval \([0, b_0]\). Since
we deduce that if \(\overline{b}\ge \sqrt{2M b_0}\), then
3 Convergence
Problems (1) and (3) are equivalent, because they are related by a non-singular change of variables. Hence, we automatically have solutions to (3). In this section the dependence of solutions on their data, \((W_0, b_0)\), plays an important role. Thus, we will write \((W,b) = (W(\cdot , \cdot , (W_0, b_0)), b(\cdot , (W_0, b_0)))\) to emphasize this dependence.
In this section, the pair \((\underline{W}_\lambda ,\underline{b}_\lambda )\) is the lower solution for a fixed \(\lambda \) given by (13) and \((\overline{W},\overline{b})\) is the upper solution given in (14). We shall write
and
to denote solutions of (3) with initial conditions \((\underline{W}_\lambda ,\underline{b}_\lambda )\) and \((\overline{W},\overline{b})\).
Lemma 3
Positivity and boundedness
There holds:
and
Proof
Repeatedly apply the comparison principle Theorem 2. \(\square \)
Lemma 4
[3] Monotonicity in time
-
(a)
The functions \(\underline{W}(\eta ,\tau )\) and \(\underline{b}(\tau )\), are non-decreasing in time.
-
(b)
The functions \({{\overline{W}}}(\eta ,\tau )\) and \(\overline{b}(\tau )\), are non-increasing in time.
Proof
We only prove part a). From Theorem 2 we deduce that
Now, for a fixed \(s=\sigma \), we consider the pair \((W^\sigma ,b^\sigma )\) where
In particular we have \(W^\sigma (\eta )\ge \underline{W}_\lambda (\eta )\) and \(b^{\sigma }\ge \underline{b}_\lambda .\) Then we apply again Theorem 2 to deduce that for every \( \tau \ge 0\)
Returning to (17), now consider \(s=\tau +\sigma \) for \(\tau \ge 0\). It holds that the pair \((W(\eta ,\tau +\sigma ,(\underline{W}_\lambda ,\underline{b}_\lambda )),b(\tau +\sigma ,(\underline{W}_\lambda ,\underline{b}_\lambda )))\) is a solution to problem (3) for the initial conditions (18) for every \(\tau > 0\). From the uniqueness of the solution we deduce that for all \(\tau \ge 0\) we have
Substituting (20) in (19) we deduce that
which completes the proof of part a). \(\square \)
We remark that if \((\underline{W},\underline{b})\) (resp. \(({{\overline{W}}}, \overline{b})\)) is defined in (15) (resp. (16)), then the Lemmas 3 and 4 imply that for every \(\lambda >0\)
In addition, the Lemmas 3 and 4 imply the convergence of \(\underline{W}\) and \({{\overline{W}}}\), namely
and
Finally we state and prove the main result of this paper, which in turn implies the result of Theorem 1.
Theorem 3
Let \((W(\eta ,\tau ,(u_0,b_0)), b(\tau ,(u_0,b_0)))\) be the solution to the free boundary problem (3) associated to the initial data \((u_0,b_0)\). If \(({U, \omega } )\) is the unique steady state of (3) given by Lemma 1, then
(a) \(\displaystyle {\lim _{\tau \rightarrow \infty } b(\tau ) = \omega }\),
(b) \(W(\cdot , \tau )\) converges to U uniformly on \([0,\beta ]\) for any \(\beta \in (0, \omega )\), when \(\tau \rightarrow \infty \).
(c) At each time \(\tau \) let us extend W to \((0, \infty )\) by the formula
Then, \({\tilde{W}}(\cdot , \tau )\) converges uniformly to \({\tilde{W}}^\infty \) on \([0,\infty )\), where
Proof
Step 1. We have to identify the limits \(\underline{W}_\infty \) and \(\overline{W}^\infty \), and improve the convergence. For this purpose, we shall show the estimate,
where \(W= \underline{W}\) or \(W={{\overline{W}}}\) are the functions defined in (15) and (16), respectively.
Proposition 1 ensures existence and uniqueness of classical solutions. When we want to restrict our attention to \(\Omega _{T,1}:=\{(\eta ,\tau ): \eta \in (0, b(\tau )), \tau \in (T, T+1)\}\), then these solutions are understood as continous up the boundary of \(\Omega _{T,1}\) with the additional following properties \(W_\eta \in C({\overline{\Omega }}_{T,1})\) and \(W_\tau , W_{\eta \eta }\in C( \Omega _{T,1})\), but this does not guarentee that \(W_{\eta \eta }\) and \(W_\tau \) are square integrable over \(\Omega _{T,1}\). This is why we set \(\psi ^\delta \in W^{1,\infty }(\{(\eta ,\tau ): \eta \in (0, b(\tau )), \tau \in (0, \infty )\}) \)
and \(\phi ^\delta \in W^{1,\infty }((0, \infty ))\), where
Finally, we set \(\varphi ^\delta (\eta ,\tau )= \psi ^\delta (\eta ,\tau )\phi ^\delta (\tau )\). Now, let us multiply the equation (3)\(-(i)\) by \(\varphi ^\delta W\) and integrate over \(\Omega _{T,1}\). 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 \(\varphi ^\delta \) does not intersect \(\partial \Omega _{T,1}.\) Now, we want to compute the limit \(L_1(T) =\lim \limits _{\delta \rightarrow 0^+} L_1^\delta (T) \). 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 \(R^\delta _1(T)\). We integrate by parts to obtain
Again here, the boundary terms vanish, because the support of \(\varphi ^\delta \) is contained in \(\Omega _{T,1}\). 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 in (24) \(M_1\) 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 \(\tau _n', \tau _n''\in [n, n+1)\) such that
Since \({{\overline{b}}}^\infty \le \overline{b}(\tau _n'')\), it follows that (25) implies the bound
Moreover, if we fix any \(\beta \in (0,{{\underline{b}}}^\infty )\), then for all sufficiently large n we have \({\underline{b}}(\tau _n')\in (\beta ,{{\underline{b}}}^\infty )\). Hence, (25) implies the bound
for all \(\beta \in (0,{{\underline{b}}}^\infty )\). Hence, we can select subsequences (not relabelled) such that
Due to (22) and (23) the limits of \({\underline{W}}(\cdot , \tau _n')\) and \({\overline{W}}(\cdot , \tau _n'')\) are uniquely defined, so is the case for \({\underline{W}}_\eta (\cdot , \tau _n')\) and \({\overline{W}}_\eta (\cdot , \tau _n'')\). Hence, we deduce that \(\psi ^{\beta }\) and \(\Psi \) do not depend on the choice of the sequence \(\tau _n\) and using (22) and (23) we conclude that \(\psi ^{\beta }\) is the weak derivative \(\underline{ W}_\eta ^\infty \) in every interval \((0,\beta )\) and that \(\Psi = {\overline{W}}_\eta ^\infty \). So that \({\underline{W}}^\infty {\chi _{(0,\beta )}}\in H^1(0,\beta )\) for all \(\beta \in (0, {{\underline{b}}}^\infty )\) and thus \({\overline{W}}^\infty \in H^1(0, \overline{b}^\infty )\). In fact, \({\underline{W}}^\infty \in H^1(0, {{\underline{b}}}^\infty )\). Indeed, in view of Lebesgue monotone convergence theorem we have
Step 3. We claim that \((\underline{W}^\infty , {\underline{b}}^\infty )\) and \((\overline{W}^\infty , {{\overline{b}}}^\infty )\) are both stationary solutions, namely solutions of (4). Hence they are smooth and equal. Indeed, we multiply the equation (3\(-i\)) by \(\varphi \in C^\infty _c(\mathbb {R})\) such that \(\varphi _\eta (0)=0\) and we integrate on \(\Omega _{T,1}\). Proceeding as in step 2 we set
where \(W= {\underline{W}}\) or \(W = {\overline{W}}\). We then deduce from Lebesgue’s dominated convergence theorem that \(\lim _{T\rightarrow \infty } L_2(T) =0.\)
Next, we investigate the right-hand-side \(R_2(T)\). Integration by parts yields
Now, we pass to the limit as \(T \rightarrow \infty \). It follows from Lebesgue’s dominated convergence theorem that
where \((W^\infty , b^\infty )\) is either \((\underline{W}^\infty ,{{\underline{b}}}^\infty )\) or \((\overline{W}^\infty , {{\overline{b}}}^\infty )\). Let us denote by \(\Phi \) an antiderivative of \(\varphi \). Then,
In addition,
Finally, we collect all the results concerning \(R_2(T)\), while keeping in mind that \(\lim _{T\rightarrow \infty }L_2(T)=0\). This yields
for all smooth functions \(\varphi \) in \(\mathbb {R}\) such that \(\varphi _\eta (0)=0\). In particular \(W^\infty \) satisfies the differential equation (4\(-i\)) in the sense of distributions.
Step 4. We recall that \(W^\infty \in H^1(0,b^\infty )\). It is easy to infer from (26) that \(W^\infty _{\eta \eta } = - \displaystyle {\frac{\eta }{2}} W^\infty _\eta \in L^2(0, b^\infty )\), which in turn implies that \(W^\infty \in H^2(0,b^\infty )\).
Next we search for the boundary condition and the conditions on the moving boundary satisfied by \(W^\infty \). After integrating by parts twice in (26) we obtain,
so that
for all smooth functions \(\varphi \) on \(\mathbb {R}\) such that \(\varphi _\eta (0)=0\). Now, if we additionally choose \(\varphi \) such that \(\varphi (b^\infty ) = \varphi _\eta (b^\infty )=0\), then (27) reduces to
and since \(\varphi (0)\) is arbitrary, we deduce that
Thus (27) becomes
Next we suppose that \(\varphi (b^\infty ) =0\), but \(\varphi _\eta (b^\infty ) \ne 0\). Then
and hence
Then, (28) becomes
Suppose that \(\varphi (b^\infty ) \ne 0\). Then (29) implies that
We deduce that the solution pair \((W^\infty , b^\infty )\) coincides with the unique solution of Problem (4), i.e. \((W^\infty , b^\infty )=(U,\omega )\) 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, \(W^\infty \in H^2(0,b^\infty )\subset C^{1, \frac{1}{2}}([0,b^\infty ])\). Moreover, the convergence of \({\overline{W}}\) to \(W^\infty \) (resp. \({\tilde{{\underline{W}}}}\) to \(W^\infty \)) is monotone on \([0,b^\infty ]\). Hence, we deduce with the help of Dini’s Theorem that this convergence is uniform.
Finally, since \(b_\lambda \le b_0 \le \overline{b}\) and since \({\underline{W}}_\lambda \le u_0\le {{\overline{W}}}\), the comparison principle implies that \({{\underline{b}}}(\tau ) \le b(\tau ) \le {\overline{b}}(\tau )\) and \({\tilde{{\underline{W}}} (\eta ,\tau )\le {\tilde{W}}(\eta ,\tau )}\le W (\eta ,\tau )\le {\overline{W}}(\eta ,\tau )\) for all \((\eta , \tau )\). We conclude that \(b(\tau ) \rightarrow b^\infty \) and that \({{\tilde{W}}}(\tau )\) converges to \(W^\infty \) uniformly on compact sets of \([0, \infty )\) as \(\tau \rightarrow \infty \). \(\square \)
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Acknowledgements
The second author, SR, was partly supported by the Project INV-006-00030 from Universidad Austral and PICT-2021-I-INVI-00317. The work of the third author, PR, was in part supported by the National Science Centre, Poland, through the Grant Number 2017/26/M/ST1/00700. A part of the work was performed while DH and SR visited the University of Warsaw, whose hospitality is greatly appreciated. The authors thank the referees for their comments which helped to improve the paper.
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Hilhorst, D., Roscani, S. & Rybka, P. Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile. Nonlinear Differ. Equ. Appl. 31, 56 (2024). https://doi.org/10.1007/s00030-024-00950-7
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DOI: https://doi.org/10.1007/s00030-024-00950-7