MATHEMATICS

I. V. BOCHAROVA

ON THE ASYMPTOTICS OF SOLUTIONS OF A FREE-BOUNDARY PROBLEM FOR THE HEAT EQUATION

(Presented by Academician I. G. Petrovskii, 3 XI 1961)

In this note we consider the asymptotic behavior, as \(t \to \infty\), of the solution of the following problem. In the domain \(D\{0 \le x \le s(t),\, 0 \le t \le T\}\), where \(s(t)\) is an unknown function, find a solution of the equation

\[ \frac{\partial^2 u}{\partial x^2}=\frac{\partial u}{\partial t}, \tag{1} \]

which satisfies the following boundary conditions:

\[ u\big|_{x=0}=f_1(t),\qquad u\big|_{x=s(t)}=f_2(t),\qquad \frac{\partial u}{\partial x}\bigg|_{x=s(t)}=g(t). \tag{2} \]

A solution of such a problem is a pair of functions \(u(x,t)\), \(s(t)\), of which \(u(x,t)\) satisfies equation (1) in the domain \(D\), and conditions (2) are fulfilled.

Under certain restrictions on the functions \(f_1(t)\), \(f_2(t)\), \(g(t)\), the existence and uniqueness of the solution of this problem were proved by T. D. Venttsel’ \((^1)\). Analogous problems were considered in the seminar of O. A. Oleinik. Similar problems arise in solving filtration problems taking bound water into account (see \((^2)\)).

Theorem. Let the functions \(f_1\), \(f_2\), \(g\) satisfy the conditions

\[ \begin{aligned} |f_1(t)-a_1|&\le \varepsilon(t),\\ |f_2(t)-a_2|&\le \varepsilon(t),\\ |g(t)-b|&\le \varepsilon(t), \end{aligned} \tag{3} \]

where the continuous function \(\varepsilon(t)\to 0\) as \(t\to\infty\); \(a_1\), \(a_2\), \(b\) are certain constants, with \(a_2>a_1\), \(b>0\). Suppose that there exists a solution \(u(x,t)\), \(s(t)\) of problem (1), (2), and that

\[ s(t)\le \frac{a_2-a_1}{b}. \]

(In the case when the existence of a solution of problem (1), (2) has been proved, the function \(s(t)\) satisfies this condition.)

Then

\[ |u(x,t)-bx-a_1|\le M_1\psi(t); \tag{4} \]

\[ \left|\frac{a_2-a_1}{b}-s(t)\right|\le M_2\psi(t), \tag{5} \]

where

\[ \psi(t)=ce^{-ct}\left(\int_0^t e^{cz}\sup_{\tau\ge z}|\varepsilon(\tau)|\,dz+\frac{1}{c}\sup_{\tau\ge 0}|\varepsilon(\tau)|\right); \]

\(M_1\), \(M_2\), \(c\) are certain positive constants.

Proof. Consider the function \(v(x,t)=u(x,t)-bx-a_1\), which satisfies the heat equation. Make the substitution \(v=\sin(k_1x+k_2)\psi(t)w(x,t)\), where \(k_2\) is an arbitrary positive constant, and

\[ k_1<\left(\frac{\pi}{2}-k_2\right)/s,\qquad s=\frac{a_2-a_1}{b}. \]

Put

\[ y(t)=\sup_{\tau\ge t}|\varepsilon(\tau)|. \]

It is clear that \(y(t)\to 0\) as \(t\to\infty\), and \(y(t)>0\) for all \(t\).

Let

\[ \psi(t)=ce^{-ct}\left(\int_0^t e^{cz}y(z)\,dz+\frac{y(0)}{c}\right), \]

where \(c=k_1^2\).

The function \(\psi(t)\) has the following properties:

1) \(\psi(t)>0\) for all \(t\);

2) \(\psi(t)\to 0\) as \(t\to\infty\); indeed,

\[ \lim_{t\to\infty}\left[ce^{-ct}\int_0^t e^{cz}y(z)\,dz+y(0)e^{-ct}\right]= \]

\[ =\lim_{t\to\infty}\frac{ce^{ct}y(t)}{e^{ct}}+\lim_{t\to\infty}y(0)e^{-ct}=0; \]

3) \(\psi(t)>y(t)\) for all \(t\); indeed,

\[ \psi(t)=ce^{-ct}\left\{\left[\frac{1}{c}y(z)e^{cz}\right]_0^t-\frac{1}{c}\int_0^t e^{cz}\,dy+\frac{y(0)}{c}\right\}= \]

\[ =ce^{-ct}\left(-\frac{y(0)}{c}+\frac{1}{c}y(t)e^{ct}-\frac{1}{c}\int_0^t e^{cz}\,dy+\frac{y(0)}{c}\right), \]

\[ \frac{\psi(t)}{y(t)}=1-\frac{e^{-ct}}{y(t)}\int_0^t e^{cz}\,dy>1, \]

since \(y(t)\) is a nonincreasing function;

4) \(\left|\dfrac{\psi'(t)}{\psi(t)}\right|\leqslant c\); we have

\[ \frac{\psi'}{\psi}=-c\left(1-\frac{y}{\psi}\right) \]

and, since

\[ \frac{y}{\psi}<1, \]

then

\[ \left|\frac{\psi'}{\psi}\right|\leqslant c. \]

The function \(w(x,t)\) satisfies the equation

\[ w_{xx}+2k_1\ctg(k_1x+k_2)w_x-\left(k_1^2+\frac{\psi'}{\psi}\right)w=w_t; \tag{6} \]

here \(k_1s+k_2<\dfrac{\pi}{2}\) and \(\sin(k_1s+k_2)\ne0\), \(\cos(k_1s+k_2)\ne0\). In view of the inequality

\[ -k_1^2-\frac{\psi'}{\psi}\leqslant 0, \]

the maximum principle holds for equation (6).

Let us estimate the function

\[ w=\frac{v}{\sin(k_1x+k_2)\psi(t)} \]

on the boundary of the domain \(D\). We have

\[ |w||_{x=0}=\left|\frac{f_1(t)-a_1}{\sin k_2\,\psi(t)}\right| \leqslant \frac{\varepsilon(t)}{\sin k_2\,\psi(t)} \leqslant M_3, \]

since

\[ \psi(t)\geqslant|\varepsilon(t)|. \]

For \(x=s(t)\),

\[ w|_{x=s(t)} = \frac{v_x|_{x=s(t)}}{\psi(t)k_1\cos(k_1s+k_2)} - \frac{w_x|_{x=s(t)}}{k_1\ctg(k_1s+k_2)}. \]

If \(\max |w(x,t)|\) is attained at \(\bar{x}=s(\bar{t})\), then at the point of a positive maximum of the function \(w(x,t)\) we have \(w_x\geqslant0,\ w>0\); hence

\[ w(x,t)|_{x=s(t)} \leqslant \frac{v_x|_{x=s(\bar{t})}}{\psi(\bar{t})\cos(k_1s+k_2)}; \]

the case of a negative minimum is treated analogously.

Therefore

\[ \left. |w| \right|_{x=s(t)} \leq \left| \frac{\left. v_x \right|_{x=s(\bar t)}}{k_1 \psi(\bar t)\cos(k_1s+k_2)} \right| \leq \left| \frac{\varepsilon(\bar t)}{k_1\psi(\bar t)\cos(k_1s+k_2)} \right| \leq M_4 . \]

Since the function \(w(x,t)\) satisfies equation (6), for which the maximum principle holds, we have \(|w(x,t)| \leq M_5\), where \(M_5=\max(M_3,M_4)\). Then

\[ |v(x,t)| \leq \sin(k_1s+k_2)\psi(t) M_5 \leq M_1\psi(t), \tag{7} \]

i.e.

\[ |u(x,t)-bx-a_1| \leq M_1\psi(t). \]

We shall show that the function \(s(t)\to \dfrac{a_2-a_1}{b}\) as \(t\to\infty\). We have

\[ \left. v \right|_{x=s(t)}=f_2(t)-a_2+a_2-bs(t)-a_1. \tag{8} \]

From relation (7) it follows that

\[ \left. |v(x,t)| \right|_{x=s(t)} \leq M_1\psi(t). \tag{9} \]

From (8) and (9) we obtain

\[ |a_2-a_1-bs(t)| \leq |M_1\psi(t)|+|a_2-f_2(t)|;\qquad \left|\frac{a_2-a_1}{b}-s(t)\right|\leq M_2\psi(t), \]

since \(|\varepsilon(t)|\leq \psi(t)\).

Received
17 X 1961

References

1. T. D. Venttsel, DAN, 131, No. 5 (1960).
2. V. A. Florin, Izv. AN SSSR, OTN, No. 11, 1625 (1951).