Reports of the Academy of Sciences of the USSR

1. Volume 148, No. 4

MATHEMATICS

A. I. Guseinov, K. K. Gasanov

ON THE APPLICABILITY OF THE FOURIER METHOD TO THE SOLUTION OF A MIXED PROBLEM FOR ONE CLASS OF QUASILINEAR HYPERBOLIC EQUATIONS

(Presented by Academician I. N. Vekua on 1 VIII 1962)

In this note various solutions of the mixed problem for a quasilinear hyperbolic equation of the form

\[ \frac{\partial^2 u}{\partial t^2} = Lu + f(\lambda, t, x, u, u_t, u_{x_1}, \ldots, u_{x_n}) \tag{1} \]

are investigated, under the initial conditions

\[ u\big|_{t=0}=\varphi(x), \qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=\psi(x) \tag{2} \]

and the boundary condition

\[ u\big|_{S}=0. \tag{3} \]

Here \(\Omega\) is an arbitrary \(n\)-dimensional domain of points \(x=(x_1,x_2,\ldots,x_n)\); \(S\) is the boundary of this domain; \(Q_l=\Omega\times[0\leq t\leq l]\) is an \((n+1)\)-dimensional cylinder; \(\lambda\) is a parameter; \(L\) is a linear self-adjoint operator

\[ Lu \equiv \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right) -a(x)u, \tag{4} \]

whose coefficients satisfy in \(\Omega\) the conditions:

\[ a(x)\geq 0, \qquad a_{ij}(x)=a_{ji}(x); \]

\[ \sum_{i,j=1}^{n} a_{ij}(x)\xi_i\xi_j \geq \alpha \sum_{i=1}^{n}\xi_i^2, \qquad \alpha=\mathrm{const}>0, \tag{5} \]

where the initial functions \(\varphi(x)\) and \(\psi(x)\) are prescribed in the domain \(\Omega\), while the function \(f(\lambda,t,x,u_0,\ldots,u_{n+1})\) is prescribed in the domain \(G_k=Q_l\times[-k<u_0,\ldots,u_{n+1}<k]\) for \(|\lambda-\lambda_0|<d\).

First, the following countable system of nonlinear integro-differential equations is studied:

\[ u_s(t)=c_s(t)+ \]

\[ +\frac{1}{\lambda_s}\int_{0}^{t}\int_{\Omega} f[\lambda,\tau,x,u(\tau,x),\ldots,u_{x_n}(\tau,x)] v_s(x)\sin\lambda_s(t-\tau)\,dx\,d\tau, \tag{6} \]

where

\[ c_s(t)=\varphi_s\cos\lambda_s t+\psi_s\sin\lambda_s t, \qquad \varphi_s=\int_{\Omega}\varphi(x)v_s(x)\,dx, \tag{7} \]

\[ \psi_s=\frac{1}{\lambda_s}\int_{\Omega}\psi(x)v_s(x)\,dx, \qquad u(t,x)=\sum_{s=1}^{\infty}u_s(t)v_s(x), \]

\(\{v_s(x)\}\) are the eigenfunctions, and \(\{\lambda_s^2\}\) are the eigenvalues of the operator \(L\).

To investigate system (6) with the aid of functions continuously differentiable on \([0,l]\), we construct all possible sequences

\(u(t)=\{u_s(t)\}\), satisfying the condition:
\[ \sum_{s=1}^{\infty}\left\{ \left[\lambda_s^\alpha \max_{0\leq t\leq l}|u_s(t)|\right]^p+ \left[\lambda_s^\beta \max_{0\leq t\leq l}|\dot u_s(t)|\right]^p\right\}<+\infty, \]
where \(\alpha\geq 0,\ \beta\geq 0,\ p\geq 1\). We shall denote this set by \(B_p^{(\alpha,\beta)}(0,l)\) and define the norm
\[ \|u(t)\|=\left(\sum_{s=1}^{\infty}\left[\lambda_s^\alpha \max_{0\leq t\leq l}|u_s(t)|\right]^p\right)^{1/p} + \left(\sum_{s=1}^{\infty}\left[\lambda_s^\beta \max_{0\leq t\leq l}|\dot u_s(t)|\right]^p\right)^{1/p}; \]
\(B_p^{(\alpha,\beta)}(0,l)\) will be a Banach-type space.

1. Generalized solution. Here two theorems are given on the existence and uniqueness of a generalized solution of problem (1), (2), (3) in the sense of paper \((3^a)\). In addition, the continuous dependence of the solution on the initial functions \(\varphi,\psi\) and the function \(f\) is studied.

For the existence and uniqueness of a generalized solution the following holds:

Theorem 1. Suppose:

a) \(\Omega\) is an arbitrary \(n\)-dimensional bounded connected domain; the coefficients of the operator \(L\) are measurable, bounded on \(\Omega\), and satisfy conditions (5);

b) \(\varphi(x)\in \overset{0}{D}(\Omega)\) *, \(\psi(x)\in L_2(\Omega)\);

c) \(f(\lambda,t,x,0,\ldots,0)\in L_2(Q_l)\), the function \(f\) is measurable in \((t,x)\) for all values of \(\lambda,u_0,\ldots,u_{n+1}\) and satisfies in \(G_\infty\) the condition
\[ \left|f(\lambda,t,x,u_0,\ldots,u_{n+1})-f(\lambda,t,x,\tilde u_0,\ldots,\tilde u_{n+1})\right| \leq \mu(t)\sum_{i=0}^{n+1}|u_i-\tilde u_i|, \tag{8} \]
where \(\mu(t)\in L_2(0,l)\).

Then problem (1), (2), (3) has a unique generalized solution.

The theorem is proved analogously to Theorems 1 and 2 of paper \((3^a)\), in contrast to which here the existence and uniqueness of the solution of system (6) in \(B_2^{(1,0)}(0,l)\) is proved.

The following theorems are more general theorems on the existence of a generalized solution of problem (1), (2), (3).

Theorem 2. Suppose conditions a), b) of Theorem 1 are satisfied. Then, if the function \(f\) satisfies the Carathéodory conditions and in the domain \(G_\infty\) the condition
\[ |f(\lambda,t,x,u_0,\ldots,u_{n+1})|\leq \mu(t)\sum_{i=0}^{n+1}|u_i|+b(t,x), \tag{9} \]
where \(\mu(t)\in L_2(0,l)\), \(b(t,x)\in L_2(Q_l)\), then problem (1), (2), (3) has at least one generalized solution.

The following theorem holds on the continuous dependence of the solution of problem (1), (2), (3) on the initial functions and the function \(f\).

Theorem 3. Let the function \(\tilde u(t,x)\) be a solution of the mixed problem
\[ \frac{\partial^2 u}{\partial t^2}=Lu+f(\lambda,t,x,u,u_t,u_{x_1},\ldots,u_{x_n})+\tilde f_0(t,x), \tag{10} \]
\[ u\big|_{t=0}=\varphi(x)+\tilde\varphi(x),\qquad \left.\frac{\partial u}{\partial t}\right|_{t=0}=\psi(x)+\tilde\psi(x),\qquad u\big|_S=0. \]

* The definition of all classes used may be found, for example, in \((3^в)\).

If the conditions of Theorem 1 are satisfied, and the norms of the functions \(\widetilde\varphi(x)\), \(\widetilde\psi(x)\), and \(\widetilde f_0(t,x)\), respectively in the sense of \(W_2^{(1)}(\Omega)\), \(L_2(\Omega)\), \(L_2(Q_l)\), differ little from zero, then the solutions of problems (1), (2), (3) and (10) differ little in the sense of \(W_2^{(1)}(\Omega,t)\).

The proof of the theorem follows from the inequality

\[ \|u(t,x)-\widetilde u(t,x)\|_{W_2^{(1)}(\Omega,t)}^2 \leq \]

\[ \leq c\left\{\|\widetilde\varphi(x)\|_{W_2^{(1)}(\Omega)}^2 +\|\widetilde\psi(x)\|_{L_2(\Omega)}^2 +\|\widetilde f_0(t,x)\|_{L_2(Q_l)}^2\right\} \exp\left(c\int_0^t \mu^2(\tau)\,d\tau\right), \]

where \(c\) is a certain constant number.

2. Solution almost everywhere. By a solution almost everywhere of problem (1), (2), (3) we shall understand it in the same sense as in [36]. We formulate two theorems on the existence and uniqueness of a solution almost everywhere. The first theorem is local in character, i.e., it holds for sufficiently small \(|\lambda-\lambda_0|\) or small \(l\); the second theorem is nonlocal in character.

Theorem 4. Suppose:

1) \(\Omega\) is an arbitrary normal three-dimensional domain contained, together with its boundary \(S\), in some open domain \(C\); the coefficients of the operator \(L\) belong to the classes

\[ a_{ij}(x)\in C^{(1,\mu)},\qquad a(x)\in C^{(0,\mu)}\qquad (\mu>0) \tag{11} \]

and satisfy conditions (5) in \(C\);

2) the initial functions \(\varphi(x)\in W_2^{(2)}(\Omega)\), \(\varphi(x),\psi(x)\in \overset{\circ}{D}(\Omega)\);

3) the function \(f(\lambda,t,x,u_0,\ldots,u_4)\) has partial derivatives with respect to \(x_i\) and with respect to \(u_i\), and in the domain \(G_k\) satisfies the conditions:

\[ \left|f'_{x_i}(\lambda,t,x,u_0,\ldots,u_4) -\widetilde f'_{x_i}(\lambda,t,x,\widetilde u_0,\ldots,\widetilde u_4)\right|\leq \]

\[ \leq b_i(\lambda,t,x)|u_0-\widetilde u_0| +b_i(\lambda,t)\sum_{j=1}^{4}|u_j-\widetilde u_j|, \]

\[ \left|f'_{u_i}(\lambda,t,x,u_0,\ldots,u_4) -f'_{u_i}(\lambda,t,x,\widetilde u_0,\ldots,\widetilde u_4)\right| \leq b(\lambda,t)|u_0-\widetilde u_0|, \]

where \(b(\lambda,t)\), \(b_i(\lambda,t)\), \(b_i(\lambda,t,x)\) belong to \(L_2\) for \(|\lambda-\lambda_0|<d\), are continuous in \(L_2\) with respect to the parameter \(\lambda\) in some neighborhood of \(\lambda_0\), and tend to zero as \(\lambda\to\lambda_0\); \(f'_{u_i}(\lambda,t,x,0,\ldots,0)\to0\) as \(\lambda\to\lambda_0\),

\[ \sup_{x\in\Omega}\left[f'_{u_i}(\lambda,t,x,0,\ldots,0)\right]^2\in L(0,l); \]

4) \(f(\lambda,t,x,0,0,u_2,u_3,u_4)\in \overset{\circ}{D}_1(Q_l)\).

Then, for sufficiently small values of \(l\) for any \(\lambda\), or for sufficiently small \(|\lambda-\lambda_0|\), there exists a unique solution almost everywhere of problem (1), (2), (3).

To prove this theorem it is first necessary to prove the existence of a solution of system (6) in \(B_2^{(2,1)}(0,l)\).

Theorem 5. Suppose the domain \(\Omega\), the coefficients of the operator \(L\), and the initial functions \(\varphi(x)\), \(\psi(x)\) satisfy the same requirements as in Theorem 4. If, for each fixed \(\lambda,u_2,u_3,u_4\), the function

\[ f(\lambda,t,x,0,0,u_2,u_3,u_4) \]

belongs to the class \(\overset{\circ}{D}_1(Q_l)\), and the function \(f\) has partial derivatives with respect to \(x_i\) and with respect to \(u_i\), satisfying in the domain \(G_\infty\) the conditions

\[ |f(\lambda,t,x,u_0,\ldots,u_4)| \leq b(\lambda,t,x)|u_0| +b(\lambda,t)\sum_{i=1}^{4}|u_i| +\widetilde b(\lambda,t,x); \]

\[ |f'_{x_i}(\lambda,t,x,u_0,\ldots,u_4)| \le b_i(\lambda,t,x)|u_0|+b_i(\lambda,t)\sum_{j=1}^{4}|u_j|+\widetilde b_i(\lambda,t,x), \]

\[ (i=1,2,3); \]

\[ |f'_{u_i}(\lambda,t,x,u_0,\ldots,u_4)| \le c_i(\lambda,t)|u_0|+\widetilde c_i(\lambda,t) \qquad (i=0,1,2,3,4), \]

where \(b(\lambda,t)\), \(b_i(\lambda,t)\), \(c_i(\lambda,t)\), \(\widetilde c_i(\lambda,t)\in L_2(0,l)\); \(b(\lambda,t,x)\), \(\widetilde b(\lambda,t,x)\), \(b_i(\lambda,t,x)\in L_2(Q_l)\) for \(|\lambda-\lambda_0|<d\), then there exists at least one solution almost everywhere of problem (1), (2), (3).

Remark. If for a generalized solution \(u(t,x)\) the function
\[ g(\lambda,t,x)=f[\lambda,t,x,u(t,x),\ldots,u_{x_n}(t,x)] \]
belongs to the class \(\overset{0}{D}_1(Q_l)\), then the generalized solution is at the same time a solution almost everywhere.

3. Classical solution. A classical solution of problem (1), (2), (3) is a function \(u(t,x)\) having in \(\overline Q_l\) continuous derivatives up to the second order and satisfying conditions (1), (2), (3) in the ordinary classical sense.

Suppose that the following conditions are satisfied:

1) \(\Omega\) is an arbitrary normal three-dimensional domain contained, together with its boundary \(S\), in some open domain \(C\); the coefficients of the operator \(L\) belong to classes (11) in the domain \(C\) and satisfy conditions (5) in \(C\). Moreover, let the boundary of the domain \(\Omega\) and the coefficients of the operator \(L\) satisfy such smoothness conditions that the series

\[ \sum_{s=1}^{\infty}\left[\frac{v_s(x)}{\lambda_s^2}\right]^2,\qquad \sum_{s=1}^{\infty}\left[\frac{\partial v_s(x)/\partial x_i}{\lambda_s^3}\right]^2,\qquad \sum_{s=1}^{\infty}\left[\frac{\partial^2 v_s(x)/\partial x_i\partial x_j}{\lambda_s^4}\right]^2 \tag{12} \]

converge uniformly in the closed domain \(\overline\Omega\), and the eigenfunctions \(v_s(x)\) have continuous derivatives up to the fourth order inside \(\Omega\).

2) \(\varphi(x)\in W_2^{(4)}(\Omega)\), \(\psi(x)\in W_2^{(3)}(\Omega)\), and \(\varphi(x)\), \(L\varphi(x)\), \(\psi(x)\), \(L\psi(x)\in \overset{0}{D}(\Omega)\).

3) The function \(f(\lambda,t,\xi_1,\ldots,\xi_8)\) has partial derivatives with respect to \(\xi_i\) up to the third order inclusive and in \(G_k\) satisfies the conditions

\[ \left| \frac{\partial^3 f(\lambda,t,\xi_1,\ldots,\xi_8)} {\partial \xi_1^{\alpha_1}\partial \xi_2^{\alpha_2}\cdots\partial \xi_8^{\alpha_8}} - \frac{\partial^3 f(\lambda,t,\xi_1,\xi_2,\xi_3,\widetilde\xi_4,\ldots,\widetilde\xi_8)} {\partial \xi_1^{\alpha_1}\partial \xi_2^{\alpha_2}\cdots\partial \xi_8^{\alpha_8}} \right| \le \]

\[ \le b_{\alpha_1\alpha_2\ldots\alpha_8}(\lambda,t,\xi_1,\xi_2,\xi_3) \sum_{i=4}^{8}|\xi_i-\widetilde\xi_i|, \]

where \(\alpha_1+\alpha_2+\cdots+\alpha_8=3\); \(b_{\alpha_1\alpha_2\ldots\alpha_8}(\lambda,t,\xi_1,\xi_2,\xi_3)\in L_2(Q_l)\) for \(|\lambda-\lambda_0|<d\).

4) For every \(\lambda\), \(u_2\), \(u_3\), \(u_4\), the functions
\[ f(\lambda,t,x,0,0,u_2,u_3,u_4), \]
\[ Lf(\lambda,t,x,0,0,u_2,u_3,u_4) \]
belong to the class \(\overset{0}{D}_1(Q_l)\).

Theorem 6. Let conditions 1, 2, 3, 4 be satisfied. Then, for sufficiently small values of \(l\), there exists a unique classical solution of problem (1), (2), (3).

Azerbaijan State University
named after S. M. Kirov

Received
30 VII 1961

CITED LITERATURE

1. O. A. Ladyzhenskaya, Mixed problem for a hyperbolic equation, Moscow, 1953.
2. V. A. Ilyin, Uspekhi Mat. Nauk, 15, no. 2 (92), 97 (1960).
3. K. K. Gasanov, a) Scientific Notes of Azerbaijan State University, Ser. Phys.-Math. and Chem. Sciences, No. 3, 47 (1960); b) No. 4, 29 (1960); c) No. 5, 13 (1960).