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UDC 532.72
HYDROMECHANICS
N. N. KOCHINA
ON ONE PERIODIC SOLUTION OF THE DIFFUSION EQUATION WITH A NONLINEAR BOUNDARY CONDITION
(Presented by Academician L. I. Sedov on 16 VI 1967)
In works (1–5) the problem was considered of finding a periodic solution of the one-dimensional diffusion equation \(\partial u/\partial t = D\, \partial^2 u/\partial x^2\) in the half-infinite region \(x \geqslant 0\) with the nonlinear boundary condition \(\partial u/\partial x \big|_{x=0} = F[u(0,t)]\) and with an a priori unknown period \(T\), depending on the solution. (Here \(F(u)\) is a multi-valued S-shaped function, and when the value \(u(0,t)\), increasing, reaches a fixed value \(u_+\), a jump occurs from the branch \(F_1(u) < 0\) to the branch \(F_2(u) > 0\); when \(u(0,t)\), decreasing, reaches \(u_-\), a jump occurs from \(F_2(u)\) to \(F_1(u)\).) It is shown below that, under certain assumptions, there exists at least one solution of this problem.
For the case \(F_2(u) = -F_1(-u)\), \(u_- = -u_+\), in paper (4) the solution of this problem was reduced to the integration of the nonlinear integral equation
\[ \lambda U = AU, \tag{1} \]
where
\[ AU = -\frac{1}{\sqrt{\pi}}\left\{\int_0^\tau \frac{F[U(\sigma)]\,d\sigma}{\sqrt{\tau-\sigma}} + \int_0^{1/2} F[U(\sigma)]\,Q(\tau-\sigma)\,d\sigma\right\}, \quad 0 \leqslant \tau \leqslant \frac{1}{2}, \tag{2} \]
\[ U(\tau)=u(0,t)/u_+, \quad \tau=t/T, \quad \lambda = u_+/\mu\sqrt{DT}, \]
\[ F(U)=\mu F_1(u)<0, \quad Q(z)=\sum_{n=1}^{\infty}\left\{-\frac{1}{\sqrt{\,n-\frac12+z\,}}+\frac{1}{\sqrt{\,n+z\,}}\right\}. \]
The function \(Q(z)\) is defined for \(-\frac12 \leqslant z \leqslant \frac12\), and \(Q(z)<0\),
\[ Q'(z)>0, \quad \lim_{z\to -1/2} Q(z)=\infty, \quad \int_{-1/2}^{1/2} Q(z)\,dz \]
converges.
We shall assume that, for values of \(U\) close to 1, the function \(F(U)\), defined by formula (2), admits the asymptotic representation
\[ F(U)=E+N(1-U)^r+\ldots \quad (1/2<r<1,\ E<0,\ N<0). \tag{3} \]
By virtue of the conditions imposed on the S-shaped function, the operator \(AU\) must be such that the function
\[ y(\tau)=AU, \tag{4} \]
where the operator \(U\) is given by formula (2), attains its maximum value \(m\) only for \(\tau=0\) and its minimum value \(-m\) only for \(\tau=\frac12\). It is easy to verify that from formulas (2) and (4) it follows that \(y(0)=-y(1/2)\).
This requirement \((-m \leq y(\tau) \leq m\) for \(0 \leq \tau \leq \frac12)\), as can be seen, will in any case be fulfilled if the function \(F(U)\) has the form
\[ F(U)=-F_0+\varkappa\varphi(U) \quad (F_0>0,\ \|\varphi(U)\|=1) \tag{5} \]
for sufficiently small \(\varkappa\).
We shall show that the operator \(AU\) is completely continuous in the space of continuous functions \(C\). By Arzelà’s theorem, for this it is sufficient to show that: 1) \(\lim AU_n=AU\) \((U_n\to U)\); 2) \(|y(\tau)|\le M\); 3) for any \(\varepsilon>0\) there exists a \(\delta(\varepsilon)\) such that \(|y(\tau)-y(\tau')|\le\varepsilon\) if \(|\tau-\tau'|<\delta(\varepsilon)\), where \(y\) is given by formula (4). Here \(U(\tau)\) is any continuous function such that \(\|U(\tau)\|=1\), and the value \(-1\) is attained by this function only for \(\tau=0\), and the value \(1\) only for \(\tau=1/2\).
Condition 1) is fulfilled by virtue of the continuity of the function \(F(U)\)
\[ \lim F(U_n)=F(U)\qquad (U_n\to U). \tag{6} \]
Introduce the notation
\[ Q_1(z)=\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n+z}}-\frac{1}{\sqrt{n+1/2+z}}\right) \quad \left(Q(z)=-\frac{1}{\sqrt{1/2+z}}+Q_1(z)\right). \tag{7} \]
For \(-1/2\le z\le 1/2\), \(Q_1(z)>0\), \(Q_1'(z)<0\), and the operator \(AU\), by virtue of (2) and (7), takes the form
\[ AU=-\frac{1}{\sqrt{\pi}}\left\{ \int_{0}^{\tau}\frac{F[U(\sigma)]\,d\sigma}{\sqrt{\tau-\sigma}} -\int_{0}^{1/2}\frac{F[U(\sigma)]\,d\sigma}{\sqrt{1/2+\tau-\sigma}} +\int_{0}^{1/2}F[U(\sigma)]Q_1(\tau-\sigma)\,d\sigma \right\}. \tag{8} \]
From (8) we have
\[ \|AU-AU_n\|\le \frac{1}{\sqrt{\pi}}\max |F[U(\sigma)]-F[U_n(\sigma)]| \{2\sqrt{2}+\tfrac12 Q_1(-1/2)\} \]
and consequently, by (6), condition 1) is fulfilled.
We shall show that condition 2) is also fulfilled. For every function \(y(\tau)\) we have \(\|y(\tau)\|=-y(0)=y(1/2)\), and consequently, by virtue of (2),
\[ \|y(\tau)\|= \frac{1}{\sqrt{\pi}}\int_{0}^{1/2}F[U(\sigma)]Q(-\sigma)\,d\sigma. \]
Using the inequalities \(F_{\min}\le F[U(\sigma)]\le F_{\max}\), we obtain for all \(y(\tau)\)
\[ m'\le \|y(\tau)\|\le M' \quad \left( M'=\frac{F_{\min}}{\sqrt{\pi}}\int_{0}^{1/2}Q(-\sigma)\,d\sigma,\quad m'=\frac{F_{\max}}{\sqrt{\pi}}\int_{0}^{1/2}Q(-\sigma)\,d\sigma \right). \tag{9} \]
Now consider condition 3). By virtue of (2) we have (for \(\tau>\tau'\))
\[ y(\tau)-y(\tau')=AU(\tau)-AU(\tau')= \]
\[ =-\frac{1}{\sqrt{\pi}}\left[ \int_{0}^{\tau'}F[U(\sigma)]\left[\frac{1}{\sqrt{\tau-\sigma}}-\frac{1}{\sqrt{\tau'-\sigma}}\right]d\sigma +\int_{\tau'}^{\tau}\frac{F[U(\sigma)]\,d\sigma}{\sqrt{\tau-\sigma}} \right. \]
\[ \left. +\int_{0}^{1/2}F[U(\sigma)]\left\{-\frac{1}{\sqrt{1/2+\tau-\sigma}}+\frac{1}{\sqrt{1/2+\tau'-\sigma}}\right\}d\sigma \right. \]
\[ \left. +\int_{0}^{1/2}F[U(\sigma)]\{Q_1(\tau-\sigma)-Q_1(\tau'-\sigma)\}\,d\sigma \right]. \tag{10} \]
Further,
\[ y(\tau)-y(\tau')\le \frac{1}{\sqrt{\pi}}\left[-F_{\min}\right]\left[ \int_{0}^{\tau'}\left(\frac{1}{\sqrt{\tau'-\sigma}}-\frac{1}{\sqrt{\tau-\sigma}}\right)d\sigma +2\sqrt{\tau-\sigma}\bigg|_{\tau'}^{\tau} \right. \]
\[ \left. +\int_{0}^{1/2}\left(\frac{1}{\sqrt{1/2+\tau'-\sigma}}-\frac{1}{\sqrt{1/2+\tau-\sigma}}\right)d\sigma +\{Q_1(-1/2)-Q_1(0)\}(\tau-\tau') \right]. \tag{11} \]
Thus,
\[ \left|y(\tau)-y(\tau')\right|\leq M\sqrt{\tau-\tau'},\qquad \varepsilon=M\sqrt{\delta} \]
\[ \left(M=K+\frac{L}{\sqrt{2}},\quad K=\frac{-4F_{\min}}{\sqrt{\pi}},\quad L=-\frac{F_{\min}}{\sqrt{\pi}}\,[Q_1(-1/2)-Q_1(0)]\right). \tag{12} \]
Let us now consider equation (1). As \(\sigma\) varies from 0 to \(1/2\), the function \(U\) in expression (2) varies from \(-1\) to 1; at the same time \(AU\) varies from \(-m\) to \(m\), and for the solution \(U(\tau)\) of equation (1) we have \(\lambda U(0)=-m\), \(\lambda U(1/2)=m\), or, since \(U(0)=-1\), \(U(1/2)=1\), \(\lambda=m=\|AU\|=y(1/2)\). Thus, instead of equation (1) one may consider the equation
\[ U=BU, \tag{13} \]
where the notation
\[ BU=AU/\|AU\|\qquad (\|BU\|=1), \tag{14} \]
has been introduced, i.e. the operator \(BU\) maps the sphere \(\|U\|=1\) into itself.
The operator \(BU\) is completely continuous; consequently, there exists a finite \(\varepsilon_0\)-net \(\eta_1,\eta_2,\ldots,\eta_p\), i.e. for any function \(U\) there is at least one element \(\eta_j\) such that \(\|BU-\eta_j\|<\varepsilon_0\) (where, for example, \(\varepsilon_0=5\varepsilon/m'\), \(\varepsilon\) is given by formula (12), \(m'\)—by (9)\(^{6}\)).
Now we construct a Schauder projection operator \(P_{n+1}\) of the set \(BU\) onto \(E^{n+1}\) such that
\[ \|P_{n+1}BU-BU\|<\varepsilon_0\qquad (\|U\|=1). \tag{15} \]
We define the operator \(P_{n+1}z\) by formula (7)
\[ P_{n+1}z=\sum_{j=1}^{p}\mu_j(z)\eta_j\Big/\sum_{j=1}^{p}\mu_j(z)\qquad (z\in BL), \]
\[ \mu_j(z)= \begin{cases} \varepsilon_0-\|z-\eta_j\|, & \text{if } \|z-\eta_j\|\leq \varepsilon_0,\\ 0, & \text{if } \|z-\eta_j\|\geq \varepsilon_0, \end{cases} \qquad (j=1,2,\ldots,p), \tag{16} \]
where \(\eta_j\in E^{n+1}\) are the elements of the finite \(\varepsilon_0\)-net. It is easy to see that (16) implies that
\[ \|P_{n+1}BU-BU\|<\varepsilon_0. \tag{17} \]
Let us now consider the operator
\[ \Phi_{n+1}U=P_{n+1}BU \tag{18} \]
on the intersection of \(L\) with \(E^{n+1}\), \(L\cap E^{n+1}\), where \(\|U\|=1\), \(U(0)=-1\), \(U(1/2)=1\).
Since the operator \(BU\) is completely continuous, the set \(BU\) is compact for every continuous function \(U\), including the case when \(U\in L\cap E^{n+1}\).
By Brouwer’s theorem, under a continuous mapping of a bounded closed convex body of the \((n+1)\)-dimensional Banach space \(E\) into itself there exists a fixed point \(^{(7,9)}\)
\[ \Phi_{n+1}U=U. \tag{19} \]
From formulas (19), (18), and (17) we obtain the inequality
\[ \|U-BU\|<\varepsilon_0. \tag{20} \]
Thus, the operator \(BU\) has “almost eigenvectors” on the sphere \(L\), \(\|U\|=1\). From the existence of almost eigenvectors there follows the theorem \(^{(7)}\): let \(B\) be a completely continuous operator defined on the boundary \(L\) of a bounded domain containing the zero \(\theta\) of the space \(E\).
Let \(\|BU\| \geq a > 0\) \((U \in L)\). Then the operator \(B\) has at least one eigenvector in \(L\). In the case considered by us, \(\|BU\| = 1\) by virtue of (14); hence the theorem is true.
Let \(U_0\) be the solution found of equation (13), \(U_0 = BU_0\). By virtue of (14), here \(BU_0 = AU_0/\|AU_0\|\), and it is clear that \(\lambda = \|AU_0\|\); from (2) we find the period of the self-oscillation
\[ T = u_+^2/\mu^2 D\lambda^2 . \tag{21} \]
In conclusion I offer my gratitude to V. N. Monakhov for a number of valuable comments.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
25 V 1967
REFERENCES
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