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MATHEMATICS
M. D. Ramazanov
ON A BOUNDARY-VALUE PROBLEM
(Presented by Academician S. L. Sobolev, 9 IV 1963)
Consider, in the domain \(Q_T=(0<t\leq T,\;0<x<1)\), the differential equation
\[ Pu+P_1(t,x)u=f(t,x), \tag{1} \]
where \(Pu\equiv u_t+(-1)^m u_{x^{2m+1}}\); \(P_1(t,x)u\equiv a_0(t,x)u_{x^{2m}}+\cdots+a_{2m}(t,x)u\); \(a_0(t,x)\in C_{t,x}^{1,2m}(\overline{Q}_T)\), \(a_k(t,x)\), \(k=1,\ldots,2m\), are summable, bounded in \(\overline{Q}_T\) functions; \(f(t,x)\in L^2(Q_T)\).
We shall be interested in the question of unique solvability of equation (1) under the conditions
\[ u(0,x)=0; \tag{2} \]
\[ u_{x^k}(t,0)=u_{x^l}(t,1)=0,\qquad k=0,\ldots,m;\quad l=0,\ldots,m-1. \tag{3} \]
(If condition (3) is satisfied for \(u(t,x)\), we shall say that \(u\in\Gamma\).)
A similar problem in the case \(m=1\) was considered by Cattabriga \((^1)\), who obtained existence and uniqueness theorems for this problem in some special cases.
Theorem 1. Let
\[ u(t,x)=\sum_{k=1}^{N} a_k(t)X_k(x), \tag{4} \]
where \(a_k(t)\in C^1(0,T)\), \(a_k(0)=0\), \(k=1,\ldots,N\); \(\{X_k\}_1^\infty\) is a complete \((^2)\) system of eigenfunctions and associated functions of the operator
\[ (-1)^{m+1}\frac{d^{2m+1}}{dx^{2m+1}}X(x),\qquad x\in(0,1),\quad X\in\Gamma. \]
Then there exist positive constants \(K\) and \(\sigma_0\) such that, for all \(\sigma>\sigma_0\), any \(T>0\), and any function \(u(t,x)\) of the form (4), the inequalities
\[ \|u_{x^l}\|_{L^2(Q_T)} \leq K\sigma^{-\frac{2m-l}{2m+1}} e^{\sigma T}\|Pu\|_{L^2(Q_T)}, \qquad l=0,\ldots,2m. \tag{5} \]
hold.
The constants \(K\) and \(\sigma_0\) depend neither on \(T\) nor on \(N\).
We extend the given function \(u(t,x)\) to the domain \((T\leq t<\infty,\;0\leq x\leq1)\) by a function \(z(t,x)\), which we construct as follows:
\[ z(t,x)\in C_{t,x}^{1,2m+1}\quad \text{in the domain } (T\leq t<\infty,\;0\leq x\leq1), \]
\[ Pz=0,\qquad z\in\Gamma,\qquad z(T,x)=u(T,x),\qquad |z(t,x)|\leq Ce^{\sigma_0 t} \]
with some constants \(C\) and \(\sigma_0>0\).
Then the function \(u(t,x)\in C_{t,x}^{1,2m+1}\) in the domains \((0<t\leq T,\;0\leq x\leq1)\) and \((T\leq t<\infty,\;0\leq x\leq1)\) is continuous at \(t=T\), and \(|u(t,x)|\leq C_1 e^{\sigma_0 t}\) with some constant \(C_1\).
Applying now the Laplace transform with respect to \(t\) to the identity
\[ u_t+(-1)^m u_{x^{2m+1}}=f\equiv Pu, \]
we obtain that the function
\[ v(\lambda,x)=\int_0^\infty u(t,x)e^{-\lambda t}\,dt \]
must satisfy the equation
\[ \lambda v+(-1)^m v_{x^{2m+1}}=F(\lambda,x),\qquad v\in\Gamma,\qquad F(\lambda,x)=\int_0^\infty Pu\,e^{-\lambda t}\,dt, \]
\[ \lambda=\sigma+i\gamma,\qquad \sigma>\sigma_0>0. \]
Consider the functions \(w_1(\lambda,x)=v(\lambda,x)\varphi(x)\) and \(w_2(\lambda,x)=v(\lambda,x)\varphi(1-x)\), where \(\varphi(x)\) is equal to 1 for \(x\in[0,\tfrac12]\), is equal to 0 for \(x\in[\tfrac34,1]\), \(\varphi(x)\in C^\infty(0,1)\), \(|\varphi(x)|\leqslant 1\). We note that \(v(\lambda,x)=w_1(\lambda,x)\) for \(0\leqslant x\leqslant \tfrac12\) and \(v(\lambda,x)=w_2(\lambda,x)\) for \(\tfrac12\leqslant x\leqslant 1\). Taking \(w_1(\lambda,x)\) to be equal to zero for \(1<x<\infty\), we may regard it on the half-axis \((0,\infty)\) as a solution of the equation
\[ \lambda w_1+(-1)^m\frac{d^{2m+1}}{dx^{2m+1}}w_1=h_1(\lambda,x), \tag{6} \]
where
\[ h_1(\lambda,x)=F(\lambda,x)\varphi(x)+(-1)^m\sum_{k=0}^{2m} C_{2m+1}^{k}v_{x^k}(\lambda,x)\varphi_{x^{2m+1-k}}(x), \]
and \(w_1(\lambda,x)\) satisfies the conditions
\[ w_{1x^k}(\lambda,0)=w_{1x^k}(\lambda,\infty)=0,\qquad k=0,\ldots,m;\quad l=0,\ldots,m-1. \]
Similarly, the function \(w_2(\lambda,x)\), extended by zero for \(x<0\), is regarded on the half-axis \((-\infty,1)\) as a solution of the equation
\[ \lambda w_2+(-1)^m\frac{d^{2m+1}}{dx^{2m-1}}w_2=h_2(\lambda,x)\equiv F(\lambda,x)\varphi(1-x)+ \]
\[ +\sum_{k=0}^{2m}(-1)^{m+1-k}C_{2m+1}^{k}v_{x^k}(\lambda,x)\varphi_{x^{2m+1-k}}(1-x) \tag{7} \]
under the conditions
\[ w_{2x^k}(\lambda,-\infty)=w_{2x^l}(\lambda,1)=0,\qquad k=0,\ldots,m;\quad l=0,\ldots,m-1. \]
With the aid of representations of the solutions of problems (6) and (7) in terms of the corresponding Green’s functions, one can prove the validity of the following estimates: for \(m\) odd,
\[ \|v_{x^k}\|_{(0,1/2)}=\|w_{1x^k}\|_{(0,1/2)} \leqslant C_2\sigma^{-\frac{2m-k}{2m+1}} \left(\|F\|_{(0,1)}+C_3\sigma^{-\frac{1}{2(2m+1)}}\sum_{l=0}^{2m}\|v_{x^l}\|_{(1/2,1)}\right), \tag{8} \]
\[ \|v_{x^k}\|_{(1/2,1)}=\|w_{2x^k}\|_{(1/2,1)} \leqslant C_2\sigma^{-\frac{2m-k}{2m+1}} \left(\|F\|_{(0,1)}+C_3\sum_{l=0}^{2m}\|v_{x^l}\|_{(0,1/2)}\right); \tag{9} \]
for \(m\) even,
\[ \|v_{x^k}\|_{(0,1/2)}=\|w_{1x^k}\|_{(0,1/2)} \leqslant C_2\sigma^{-\frac{2m-k}{2m+1}} \left(\|F\|_{(0,1)}+C_3\sum_{l=0}^{2m}\|v_{x^l}\|_{(1/2,1)}\right), \tag{10} \]
\[ \|v_{x^k}\|_{(1/2,1)}=\|w_{2x^k}\|_{(1/2,1)} \leqslant C_2\sigma^{-\frac{2m-k}{2m+1}} \left(\|F\|_{(0,1)}+C_3\sigma^{-\frac{1}{2(2m+1)}}\sum_{l=0}^{2m}\|v_{x^l}\|_{(0,1/2)}\right), \tag{11} \]
\[ k=0,1,\ldots,2m. \]
Here \(\|y\|_{(a,b)}\) denotes, as usual,
\[ \left(\int_a^b |y(x)|^2\,dx\right)^{1/2}. \]
Take \(\sigma\) sufficiently large and solve (8)—(9) (or (10)—(11)) with respect to \(\|v_{x^k}\|_{(0,1/2)}\), \(\|v_{x^k}\|_{(1/2,1)}\), \(k=0,\ldots,2m\). Then, for even as well,
and for odd \(m\)
\[ \left\|v_{x^{k}}\right\|_{(0,1)} \leq K\sigma^{-\frac{2m-k}{2m+1}}\|F\|_{(0,1)}, \qquad k=0,\ldots,2m . \tag{12} \]
The estimates (5) are now easily obtained with the aid of Parseval’s equality from the inequalities (12).
It follows from Theorem 1 that on the set of functions (4) one can introduce the scalar product
\[ [u,v]=(Pu,Pv)_{\mathscr L^2(Q_T)} . \tag{13} \]
The Hilbert space obtained by completing this set in the metric (13) will be denoted by \(H(Q_T)\). The inequalities (5) show that the embedding \(H(Q_T)\subset W^{0,2m}_{t,x}(Q_T)\) is valid.
Definition. A function \(u(t,x)\in H(Q_T)\) is called a generalized solution of problem (1)—(3) if there exists a sequence of functions \(u_n(t,x)\) of the form (4), converging as \(n\to\infty\) to \(u(t,x)\) in the norm of \(H(Q_T)\), and such that
\[ \lim_{n\to\infty}\left\|Pu_n+P_1(t,x)u_n-f(t,x)\right\|_{\mathscr L^2(Q_T)}=0 . \]
It is easy to see that the generalized solution thus defined satisfies the integral identity
\[ -(u,w_t)_{\mathscr L^2(Q_T)} +(-1)^{m-1}(u_{x^{2m}},w_x)_{\mathscr L^2(Q_T)} +(P_1u,w)_{\mathscr L^2(Q_T)} =(f,w)_{\mathscr L^2(Q_T)} \]
for any function \(w\in \dot W^1_2(Q_T)\), \(w(T,x)=w(t,0)=w(t,1)=0\).
Theorem 2. For every function \(f(t,x)\in \mathscr L^2(Q_T)\) there exists a unique generalized solution of problem (1)—(3).
Lemma. For every function \(f(t,x)\in \mathscr L^2(Q_T)\) there exists a unique generalized solution of the problem
\[ Pu=f,\qquad u(0,x)=0,\qquad u\in \Gamma . \]
The validity of the lemma follows from Theorem 1 and the completeness of the system of functions \(\{X_k\}_1^\infty\) in \(\mathscr L^2(0<x<1)\) \((^2)\).
Remark. After the substitution
\[ u(t,x)=v(t,x)\exp\left[(-1)^{m+1}\int_0^x a(t,\xi)\,d\xi\right] \]
for the function \(v(t,x)\) one obtains an equation of the same form as (1), but with \(a_0(t,x)=0\). Therefore it suffices to carry out the proof of Theorem 2 for the equation
\[ Pu+P_2(t,x)u=f, \tag{14} \]
\[ P_2(t,x)u=a_1(t,x)u_{x^{2m-1}}+\cdots+a_{2m}(t,x)u;\qquad a_k(t,x),\ k=1,\ldots,2m, \]
summable bounded functions in \(\overline Q_T\).
From the lemma follows the existence of the operator \(P^{-1}:\mathscr L^2(Q_T)\to H(Q_T)\), and hence also the existence of the operator \(A=P^{-1}P_2:H(Q_T)\to H(Q_T)\). In this case problem (14), (2), (3) turns out to be equivalent to the operator equation in the space \(H(Q_T)\)
\[ u+Au=h=P^{-1}f . \tag{15} \]
Using the estimates (5), one can show that \(A\) is a completely continuous operator, i.e. equation (15) is of Fredholm type. Therefore, to complete the proof of Theorem 2 it is enough to show the uniqueness of the trivial solution of the homogeneous equation
\[ Pu+P_2u=0,\qquad u(0,x)=0,\qquad u\in \Gamma . \tag{16} \]
Let the function \(u(t,x)\) be a solution of this problem in the domain \(Q_T\); then \(u(t,x)\) is also a solution in any smaller domain \(Q_{T_1}\), \(0<T_1\leq T\).
From Theorem 1 we have
\[ \|u\|_{H(Q_{T_1})}=\|Pu\|_{\mathscr L^2(Q_{T_1})} =\|P_2u\|_{\mathscr L^2(Q_{T_1})} \leq M\sum_{k=0}^{2m-1}\|u_{x^k}\|_{\mathscr L^2(Q_{T_1})}\leq \]
\[ \leq MK\sigma^{-\frac{1}{2m+1}} \sum_{k=0}^{2m-1}\sigma^{-\frac{2m-k-1}{2m+1}} e^{\sigma T_1}\|u\|_{H(Q_{T_1})}, \tag{17} \]
where \(M\) is the maximum of the moduli of the coefficients of the operator \(P_2\).
Choose \(\sigma\) so large that
\[ MK\sigma^{-\frac{1}{2m+1}} \sum_{k=0}^{2m-1}\sigma^{-\frac{2m-k-1}{2m+1}}<\frac14. \]
Then, for \(T_1<\frac{1}{\sigma}\ln 3\), inequality (17) is possible only when
\(\|u\|_{H(Q_{T_1})}=0\). That is, \(u=0\) in the domain
\((0<t<T,\;0<x<1)\).
By the definition of a generalized solution, there exists a sequence of smooth functions \(u_n(t,x)\),
\[ u_n(t,x)=\sum_{k=1}^{N_n} a_k(t)X_k(x),\qquad u_n(0,x)=0, \]
converging in \(H(Q_T)\) to the function \(u(t,x)\) and such that
\[ \lim_{n\to\infty}\|Pu_n+P_2(t,x)u_n\|_{\mathscr L^2(Q_T)}=0. \]
It can be shown that the functions \(v_n(t,x)=u_n(t,x)\psi(t)\), where
\(\psi(t)\in C^1(0,T)\), \(\psi(t)=0\) for \(0\leq t\leq T_1/2\);
\(\psi(t)=1\) for \(T_1/2\leq t\leq T_1\), \(|\psi(t)|\leq1\), form a sequence of the same type as \(\{u_n\}\). Moreover,
\(v_n(T_1/2,x)=0\).
Thus, the sequence of functions \(v_n^{(1)}(t,x)=v_n(t+T_1/2,x)\) defines in the space \(H(Q_{T-T_1/2})\) the function
\[ u^{(1)}(t,x)=\lim_{n\to\infty} v_n^{(1)}(t,x) \quad \text{(in } H(Q_{T-T_1/2})\text{),} \]
which is, in the domain \(Q_{T-T_1/2}\), a generalized solution of the problem
\[ Pu^{(1)}+P_2\left(t+\frac{T_1}{2},x\right)u^{(1)}=0, \qquad u^{(1)}(0,x)=0,\qquad u^{(1)}\in\Gamma. \]
Applying inequality (17) again, we obtain
\[ v^{(1)}(t,x)=0\qquad \text{for }0<t<T_1, \]
i.e.
\[ u(t,x)=0\qquad \text{for }0<t<\frac32 T_1. \]
Taking, instead of \(\psi(t)\), a function \(\psi_1(t)\) equal to 0 for \(0\leq t\leq T_1\) and equal to 1 for \(\frac32 T_1\leq t\leq T\), \(|\psi_1(t)|\leq1\), \(\psi_1(t)\in C^1(0,T)\), and repeating all the preceding arguments, we obtain \(u(t,x)=0\) in the domain \((0<t<2T_1,\;0<x<1)\). It is clear that in this way, after a finite number of steps, we prove that \(u=0\) in the entire domain \(Q_T\), as was required.
In conclusion, the author expresses sincere gratitude to V. I. Mikhailov for posing the problem and for supervising the work.
Moscow State University
named after M. V. Lomonosov
Received
3 IV 1963
References
- L. Cattabriga, Ann. Scuola Norm. Super. Pisa, Ser. 3, 13, No. 2, 163 (1959).
- M. V. Keldysh, DAN, 77, 11 (1951).