Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1967. Vol. 174, No. 2
UDC 517.9:533.9
MATHEMATICS
G. Ya. LYUBARSKII
AN EQUATION OF PARABOLIC TYPE WITH A PERIODIC COEFFICIENT
(Presented by Academician I. G. Petrovskii on 7 VII 1966)
The subject of the present note is the Cauchy problem for the parabolic equation
\[ \begin{gathered} \partial u/\partial t=\partial^{2}u/\partial x^{2}+q(x,\tau)u,\qquad \tau=\nu t,\\ Au(x,t)\equiv a_{1}u'(0,t)+a_{2}u(0,t)=0,\\ Bu(x,t)\equiv b_{1}u'(1,t)+b_{2}u(1,t)=0,\\ \operatorname*{l.i.m.}_{t\to0} u(x,t)=\psi(x), \end{gathered} \tag{1} \]
in the half-strip \(\Pi\) \((0\le x\le 1,\ t>0)\). The numbers \(a_1,a_2,b_1\), and \(b_2\) are real, with \(a_1^2+a_2^2>0\) and \(b_1^2+b_2^2>0\); \(\psi(x)\) is an arbitrary quadratically integrable function; \(\nu\) is a positive parameter. We shall be interested in those properties of problem (1) that it possesses for sufficiently large values of the parameter \(\nu\).
Regarding the function \(q(x,\tau)\) the following is assumed. It is real and periodic with period \(2\pi\), expands in an absolutely convergent Fourier series
\[ q(x,\tau)=\sum_{k=-\infty}^{\infty} q_k(x)e^{ik\tau} \]
with continuous coefficients, and has a quadratically integrable derivative with respect to \(t\). In addition, it is assumed that there exists a sufficiently small positive number \(\sigma\) such that the series
\[ \|q\|_\sigma=\max_{0\le x\le 1}|q_0(x)|+\sum_{k=-\infty}^{\infty}|k|^\sigma \max_{0\le x\le 1}|q_k(x)| \]
converges.
We shall agree to call a Floquet solution of problem (1) any function \(u(x,t)\) satisfying the first three relations in (1) and representable in the form
\[ u(x,t)=e^{\gamma t}v(x,t), \tag{2} \]
where \(v(x,t)\) is a periodic function with period \(T=2\pi/\nu\).
Denote by \(\Lambda_\mu\) \((0\le \mu\le 1)\) the totality of all periodic functions \(y(x,t)=y(x,t+T)\) whose Fourier coefficients are continuous and satisfy the condition: the series
\[ \|y\|_\mu=\max_{0\le x\le 1}|y_0(x)|+\sum_{k=-\infty}^{\infty}{}' |k|^\mu \max_{0\le x\le 1}|y_k(x)| \tag{3} \]
converges. If \(\|y\|_\mu\) is taken as the norm of the function \(y(x,t)\), then \(\Lambda_\mu\) becomes a Banach space.
Theorem 1. If \(\nu\) is greater than a certain number \(\nu_m(\|q\|_0)\), depending only on the magnitude of \(\|q\|_0\),
\[ \nu>\nu_m(\|q\|_0), \]
then problem (1) has at least \(m\) Floquet solutions \(u_k(x,t)\) \((k=1,2,\ldots,m)\), belonging to \(\Lambda_1\).
Denote by \(L\) the operator
\[ L=d^2/dx^2+q_0(x), \]
considered on functions satisfying the boundary conditions \(A\varphi=0,\ B\varphi=0\). Let \(\varphi_k(x)\) and \(\lambda_k\) \((k=1,2,\ldots)\) be a complete set of eigenfunctions and eigenvalues of the operator \(L\), with \(\lambda_{k+1}<\lambda_k\). The eigenfunctions \(\varphi_k(x)\) have norm equal to one in \(\mathcal L_2(0,1)\), and are uniformly bounded:
\[ \int_0^1 \varphi_k^2(x)\,dx=1,\qquad \max_{0\le x\le 1}|\varphi_k(x)|<\theta \quad (k=1,2,\ldots). \]
Theorem 2. If \(\nu>\nu_m(\|q\|_0)\) and \(\lambda_{m+1}<0\), then the solution of problem (1) can be represented in the form
\[ u(x,t)=\sum_{k=1}^m c_k u_k(x,t)+w(x,t); \tag{4} \]
here \(c_k\) \((k=1,2,\ldots,m)\) are certain coefficients, determined below, and \(w(x,t)\) is a function that can be estimated as follows:
\[ \left|\,w(x,t)-\sum_{k=m+1}^{\infty} h_k e^{\lambda_k t}\varphi_k(x)\,\right|< \tag{5} \]
\[ <2\theta\|h\|\left[1+2(\alpha-\lambda_{m+1}) \sum_{k=m+2}^{\infty}(\lambda_{m+1}-\lambda_k)^{-1}\right]^{1/2} e^{\alpha t}F_m\!\left(Q_1\sqrt{\frac{2t}{\alpha-\lambda_{m+1}}}\right); \]
\(h_k\) are the Fourier coefficients of the function
\[ h(x)=\psi(x)-\sum_{k=1}^m c_k u_k(x,0); \]
\(\alpha\) is an arbitrary number from the interval \(\lambda_{m+1}<\alpha\le 0\);
\[ q_1(x,\tau)\equiv q(x,\tau)-q_0(x),\qquad Q_1=\max_{x,\tau}|q_1(x,\tau)|; \tag{6} \]
\(\|h\|\) denotes the norm of the function \(h(x)\) in \(\mathcal L_2(0,1)\), and, finally,
\[ F_m(z)\equiv \sum_{n=m+1}^{\infty}\frac{z^n}{\sqrt{n!}}. \]
The function \(w(x,t)\) has a continuous derivative with respect to \(t\) in the half-strip \(\Pi\).
Let us note that the right-hand side of inequality (5) for \(\alpha=0\) decreases as the index \(m\) grows proportionally to \(|\lambda_m|^{-(m+1)/2}\) uniformly in \(t\) \((t\ge0)\). On the other hand, with a suitable choice of the parameter \(\alpha\), the right-hand side does not exceed the quantity
\[ M e^{(\lambda_{m+1}+2Q_1^2+\varepsilon)t} \]
for all \(t\ge0\), if \(\varepsilon>0\) is any positive number and \(M\) is a sufficiently large number.
Denote by \(z_k(x,t)\in\Lambda_1\) \((k=1,2,\ldots,m)\) the Floquet solutions of problem (1) in which the function \(q(x,t)\) is replaced by the function \(q(x,-t)\). The functions \(z_k(x,t)\) may be renumbered and normalized so that
\[ \int_0^1 u_i(x,t)z_k(x,-t)\,dx=\delta_{ik} \qquad (i,k=1,2,\ldots,m). \]
Theorem 3. The coefficients \(c_k\) \((k=1,2,\ldots,m)\) appearing in relations (4) and (5) can be computed by the formula
\[ c_k=\int_0^1 \psi(x)z_k(x,0)\,dx. \]
The following two theorems make it possible to find Floquet solutions \(u_k(x,t)\) and the function \(w(x,t)\).
Theorem 4. We represent the Floquet solution \(u_k(x,t)\) \((k=1,2,\ldots,m)\) in the form (2). The function \(v(x,t)\) is the limit in the norm \(\Lambda_1\) of the sequence \(v_n(x,t)\), defined as follows:
- \(v_0(x,t)\equiv \varphi_k(x)\).
- If the function \(v_n(x,t)\) \((n=0,1,\ldots)\) has already been defined, then the function \(v_{n+1}(x,t)\) is found as the solution in \(\Lambda_1\) of the problem
\[ \partial v_{n+1}/\partial t=(L-\lambda_k)v_{n+1}+[q_1(x,t)-\gamma_n+\lambda_k]v_n, \tag{7} \]
\[ Av_{n+1}=0,\qquad Bv_{n+1}=0,\qquad A^+v_{n+1,0}\equiv a_1v_{n+1,0}(0)-a_2v'_{n+1,0}(0)=A^+\varphi_k(x), \]
where
\[ \gamma_n=\lambda_k+ \int_0^1 \varphi_k(x)\bigl(q_1(x,t)v_n(x,t)\bigr)_0\,dx \Big/ \int_0^1 \varphi_k(x)v_{n,0}(x)\,dx. \tag{8} \]
All the numbers \(\gamma_n\) determined in this way are finite, and the sequence of these numbers tends to \(\gamma\). The rate of approximation of the function \(v(x,t)\) by the functions \(v_n(x,t)\) is characterized by the following estimates:
\[ \left|v_{n+1,0}(x)-v_{n,0}(x)\right| <N_0\left(\frac{a_0}{\nu}\right)^{1+[n/2]}, \]
\[ \left\|v_{n+1,1}(x,t)-v_{n,1}(x,t)\right\|_0 <N_1\left(\frac{a_1}{\nu}\right)^{1+\left[\frac{n+1}{2}\right]} \qquad (n=0,1,\ldots). \tag{9} \]
Here \(N_0,N_1,a_0\), and \(a_1\) are some constants independent of \(\nu\).
The indices zero and one in equalities (7), (8), and (9) have the same meaning as in definition (6).
Theorem 5. The function \(w(x,t)\) is equal to
\[ w(x,t)=w_0(x,t)+\sum_{n=1}^{\infty} w_n(x,t); \tag{10} \]
the series on the right-hand side converges uniformly in \(\Pi\), and the functions \(w_n(x,t)\) \((n=0,1,\ldots)\) are defined as follows:
\[ w_n(x,t)=\sum_{k=1}^{m}c_k^{(n)}(t)\varphi_k(x)+\xi_n(x,t) \qquad (n=0,1,\ldots), \tag{11} \]
\[ \xi_0(x,t)=\sum_{k=m+1}^{\infty} h_k e^{\lambda_k t}\varphi_k(x),\qquad \xi_n(x,t)=\sum_{k=m+1}^{\infty}\varphi_k(x)\int_0^t e^{\lambda_k(t-s)} f_{n-1,k}(s)\,ds, \]
\[ (n=1,2,\ldots), \tag{12} \]
\(f_{n,k}(s)\) are the Fourier coefficients of the function \(q_1(x,\nu s)w_n(x,s)\) with respect to the functions \(\varphi_k(x)\) \((k=1,2,\ldots)\). The coefficients \(c_k^{(n)}(t)\) are determined from the conditions
\[ \int_0^1 w_n(x,t)z_k(x,-t)\,dx=0 \qquad (t\ge 0,\ k=1,2,\ldots,m;\ n=0,1,\ldots). \tag{13} \]
The terms of the series (10) can be estimated as follows:
\[ \left| w_n(x,t) \right| \leq 2\theta \|h\| \left[ 1+2(a-\lambda_{m+1}) \sum_{k=m+2}^{\infty}(\lambda_{m+1}-\lambda_k)^{-1} \right]^{1/2} \times \]
\[ {}\times \frac{e^{at}}{\sqrt{n!}} \left[ \frac{2Q_1^2 t}{\alpha-\lambda_{m+1}} \right]^{n/2}, \tag{14} \]
where \(a\) is any number from the interval \((\lambda_{m+1}<a\leq 0)\).
The function \(w(x,t)\) is continuously differentiable with respect to \(t\) in \(\Pi\). Its derivative can be obtained by termwise differentiation of the series (10). The estimate
\[ \left|\dot w_n(x,t)\right| \leq 2\sigma_n(t_0)(t/t_0)^{3/4n}(1-t_0/t)^{-1/3} \qquad (n=1,2,\ldots), \]
holds, where \(t_0<t\) is an arbitrary positive number, and \(\sigma_n(t_0)\) is a collection of functions such that
\[ \lim_{n\to\infty}\sqrt[n]{\sigma_n(t_0)}=0. \]
Received
27 VI 1966