MATHEMATICS

A. I. GUSEINOV, K. I. KHUDAVERDIEV

ON THE SOLUTION BY THE FOURIER METHOD OF A ONE-DIMENSIONAL MIXED PROBLEM FOR QUASILINEAR HYPERBOLIC EQUATIONS OF SECOND ORDER

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

The present work is a development and continuation of the works \((^{1a,b,c})\), in which the existence and uniqueness of a generalized solution, of a solution almost everywhere, and of a classical solution of the following one-dimensional mixed problem were studied:

\[ \frac{\partial^{2}u}{\partial t^{2}}-a^{2}\frac{\partial^{2}u}{\partial x^{2}} =\lambda F\left[t,x,u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\right], \]

\[ u(t,0)=u(t,l)=0, \tag{A} \]

\[ u(0,x)=\varphi(x),\qquad u_t'(0,x)=\psi(x), \]

where \(0\leq t\leq T<\infty\); \(0\leq x\leq l<\infty\); \(a>0\); \(\lambda\) is a parameter; \(F,\varphi,\psi\) are given functions.

In the present paper we establish theorems on the existence and uniqueness of a generalized solution, of a solution almost everywhere, and of \(k\)-times \((k\geq 2)\) continuously differentiable solutions of problem (A). We formulate the main results obtained.

§ 1. Generalized solution of problem (A).

Theorem 1. Suppose:

1. The function \(\varphi(x)\) is continuous on the interval \([0,l]\), \(\varphi'(x)\in L_2[0,l]\), \(\varphi(0)=\varphi(l)=0\); \(\psi(x)\in L_2[0,l]\).

2. \(F[t,x,0,0,0]\in L_2(D)\), where \(D=(0\leq t\leq T)\times(0\leq x\leq l)\).

3. The function \(F[t,x,u,v,w]\), defined in the domain
   \[ D\times(-R<u<R)\times(-\infty<v,w<\infty), \]
   is measurable with respect to \((t,x)\) for fixed \((u,v,w)\) and, for almost all \((t,x)\in D\), satisfies the condition
   \[ \left|F[t,x,u,v,w]-F[t,x,\widetilde u,\widetilde v,\widetilde w]\right| \leq a(t,x)|u-\widetilde u|+ \]
   \[ +\,b(t)\,[\,|v-\widetilde v|+|w-\widetilde w|\,], \]
   where \(a(t,x)\in L_2(D)\), \(b(t)\in L_2[0,T]\),

\[ R> \frac{V^{2}l\,(l^{2}+a^{2}\pi^{2})}{al} \left[ a^{2}\|\varphi'(x)\|_{L_2[0,l]}^{2} +\|\psi(x)\|_{L_2[0,l]}^{2} +\right. \]
\[ \left. +\lambda^{2}T\|F[t,x,0,0,0]\|_{L_2(D)}^{2} \right]^{1/2} \times \]
\[ \times \sum_{s=0}^{\infty} \left\{ \frac{ \left( \frac{\lambda^{2}T(l^{2}+a^{2}\pi^{2})}{a^{2}\pi^{2}l^{2}} \left[ \pi^{2}l\|a(t,x)\|_{L_2(D)}^{2} +3(l^{2}+\pi^{2})\|b(t)\|_{L_2[0,T]}^{2} \right] \right)^{s} }{s!} \right\}^{1/2}. \]

Then problem (A) has a unique generalized solution satisfying the condition

\[ \max_{(t,x)\in D}|u(t,x)|<R \tag{1} \]

and depending continuously on the initial functions \(\varphi(x),\psi(x)\) and the parameter \(\lambda\) in the sense that a small change in the numbers \(\|\varphi'(x)\|_{L_2[0,l]}\), \(\|\psi(x)\|_{L_2[0,l]}\) and ...

\(\lambda\) there corresponds a change of this solution that is small in the norm of the space \(B_{1,0}^{\infty,2}\) (\(([^4\mathrm{r}])\), p. 17).

Theorem 2. Let:

1. The first and second conditions of Theorem 1 be satisfied.

2. The third condition of Theorem 1 be satisfied, where

\[ R>N\equiv \frac{1}{a}\sqrt{\frac{2}{3l}\,(l+a\pi)\left(a^2\|\varphi'(x)\|_{L_2[0,l]}^2+\|\psi(x)\|_{L_2[0,l]}^2\right)}. \]

1. The inequality

\[ \frac{2|\lambda|(l+a\pi)\sqrt{3T}}{3al\pi} \left\{ l\pi^2\|F[t,x,0,0,0]\|_{L_2(D)}^2+ [l\pi^2]\|a(t,x)\|_{L_2(D)}^2+ 3(l^2+\pi^2)\|b(t)\|_{L_2[0,T]}^2 R^2 \right\}^{1/2} < R-N. \]

Then problem (A) has a unique generalized solution satisfying condition (1) and depending continuously on the initial functions \(\varphi(x)\), \(\psi(x)\) and on the parameter \(\lambda\) in the sense indicated in Theorem 1.

Now, along with problem (A), let us consider the problem:

\[ \frac{\partial^2 u}{\partial t^2}-a^2\frac{\partial^2 u}{\partial x^2} = \widetilde{\lambda}\Phi\left[t,x,u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\right], \]

\[ u(t,0)=u(t,l)=0, \]

\[ u(0,x)=\widetilde{\varphi}(x),\qquad u_t'(0,x)=\widetilde{\psi}(x). \tag{\(\widetilde{\mathrm A}\)} \]

Theorem 3. Let:

1. The functions \(\varphi(x)\), \(\widetilde{\varphi}(x)\), \(\psi(x)\), \(\widetilde{\psi}(x)\) satisfy, respectively, condition 1 of Theorem 1.

2. The functions \(F[t,x,u,v,w]\), \(\Phi[t,x,u,v,w]\), defined in the domain \(D\times(-\infty<u,v,w<\infty)\), be measurable in \((t,x)\) for all fixed \((u,v,w)\) and, for almost all \((t,x)\in D\), satisfy the condition

\[ |F[t,x,u,v,w]-F[t,x,\widetilde{u},\widetilde{v},\widetilde{w}]| \leq a(t,x)|u-\widetilde{u}|+b(t)\bigl[|v-\widetilde{v}|+|w-\widetilde{w}|\bigr], \]

\[ |\Phi[t,x,u,v,w]-\Phi[t,x,\widetilde{u},\widetilde{v},\widetilde{w}]| \leq \widetilde{a}(t,x)|u-\widetilde{u}|+ \widetilde{b}(t)\bigl[|v-\widetilde{v}|+|w-\widetilde{w}|\bigr], \]

where \(a(t,x),\widetilde{a}(t,x)\in L_2(D)\), \(b(t),\widetilde{b}(t)\in L_2[0,T]\).

1. \(F[t,x,0,0,0]\), \(\Phi[t,x,0,0,0]\in L_2(D)\).

2. In the domain \(D\times(-R\leq u\leq R)\times(-\infty<v,w<\infty)\) one has

\[ |F[t,x,u,v,w]-\Phi[t,x,u,v,w]|\leq c(t,x), \]

where \(c(t,x)\in L_2(D)\),

\[ R^2= \frac{(l^2+a^2\pi^2)}{a^2l} \left[ a^2\|\varphi'(x)\|_{L_2[0,l]}^2+ \|\psi(x)\|_{L_2[0,l]}^2+ 4\lambda^2T\|F[t,x,0,0,0]\|_{L_2(D)}^2 \right]\times \]

\[ \times \exp\left\{ \frac{4\lambda^2T(l^2+a^2\pi^2)}{a^2\pi^2l^2} \left[ l\pi^2\|a(t,x)\|_{L_2(D)}^2+ 3(l^2+\pi^2)\|b(t)\|_{L_2[0,T]}^2 \right] \right\}. \]

Then for any \(\varepsilon>0\) one can specify such a \(\delta(\varepsilon)>0\) that

\[ \|u(t,x)-\widetilde{u}(t,x)\|_{B_{1,0}^{\infty,2}}<\varepsilon, \]

provided only that each of the quantities

\[ \|\varphi'(x)-\widetilde{\varphi}'(x)\|_{L_2[0,l]},\quad \|\psi(x)-\widetilde{\psi}(x)\|_{L_2[0,l]},\quad \|c(t,x)\|_{L_2(D)},\quad |\lambda-\widetilde{\lambda}| \]

is less than \(\delta\), where \(u(t,x)\), \(\widetilde{u}(t,x)\) are, respectively, the generalized solutions of problems (A), \((\widetilde{\mathrm A})\).

Theorem 4. If condition 3 of Theorem 1 is satisfied, where \(R\) is any positive number, then problem (A) has no more than one generalized solution satisfying condition (1).

§ 2. Solution almost everywhere of problem (A).

Theorem 5. Suppose:

1. The function \(\varphi(x)\) is continuously differentiable on the interval \([0,l]\), \(\varphi''(x)\in L_2[0,l]\), \(\varphi(0)=\varphi(l)=0\).

2. The function \(\psi(x)\) is continuous on the interval \([0,l]\), \(\psi'(x)\in L_2[0,l]\), \(\psi(0)=\psi(l)=0\).

3. The function \(F[\xi_1,\xi_2,\xi_3,\xi_4,\xi_5]\), continuous in the domain
   \(Q=D\times(-R<\xi_3,\xi_4,\xi_5<R)\) with respect to the totality of its arguments, has partial derivatives \(F'_{\xi_i}\) \((i=2,\ldots,5)\), defined in this same domain and measurable with respect to \((\xi_1,\xi_2)\) for fixed \((\xi_3,\xi_4,\xi_5)\), where

\[ R^2>\frac{2(l+a\pi)^2(\max\{l,\pi\})^2}{3a^2\pi^2l} \left(a^2\|\varphi''(x)\|^2_{L_2[0,l]}+\|\psi'(x)\|^2_{L_2[0,l]}\right). \]

1. Almost for all \((\xi_1,\xi_2)\in D\) one has

\[ \left|F'_{\xi_i}[\xi_1,\xi_2,\xi_3,\xi_4,\xi_5] - F'_{\xi_i}[\xi_1,\xi_2,\widetilde{\xi}_3,\widetilde{\xi}_4,\widetilde{\xi}_5]\right| \leq a(\xi_1,\xi_2)\sum_{s=3}^{5}|\xi_s-\widetilde{\xi}_s|, \]

\[ \left|F'_{\xi_i}[\xi_1,\xi_2,\xi_3,\xi_4,\xi_5] - F'_{\xi_j}[\xi_1,\xi_2,\widetilde{\xi}_3,\widetilde{\xi}_4,\widetilde{\xi}_5]\right| \leq b(\xi_1)\sum_{s=3}^{5}|\xi_s-\widetilde{\xi}_s|, \]

where \(a(\xi_1,\xi_2)\in L_2(D)\), \(b(\xi_1)\in L_2[0,T]\), \(i=2,3\), \(j=4,5\).

1. \(F[\xi_1,0,0,0,\xi_5]=F[\xi_1,l,0,0,\xi_5]=0\), \(F'_{\xi_i}[\xi_1,\xi_2,0,0,0]\in L_2(D)\),

\[ \sup_{0<x<l}|F'_{\xi_j}[t,x,0,0,0]|=g_j(t)\in L_2[0,T], \]

where \(i=2,3\), \(j=4,5\).

Then problem (A) has a unique solution almost everywhere, satisfying in \(D\) the condition

\[ -R<u(t,x),\ u'_t(t,x),\ u'_x(t,x)<R, \tag{2} \]

under each of the following conditions: a) one of the numbers \(T,|\lambda|\) is fixed, and the other is sufficiently small; b) \(\lambda,T\) are fixed; \(\|\varphi''(x)\|_{L_2[0,l]}\), \(\|\psi'(x)\|_{L_2[0,l]}\), \(\|F'_x[t,x,0,0,0]\|_{L_2(D)}\) are sufficiently small.

Theorem 6. If the function \(F[t,x,u,v,w]\), defined in the domain
\(D\times(-R_1<u<R_1)\times(-R_2<v<R_2)\times(-R_3<w<R_3)\), almost for all \((t,x)\in D\) satisfies the condition

\[ |f[t,x,u,v,w]-F[t,x,\widetilde{u},\widetilde{v},\widetilde{w}]| \leq a(t,x)|u-\widetilde{u}|+ \]

\[ +b(t)\bigl[|v-\widetilde{v}|+|w-\widetilde{w}|\bigr], \]

where \(a(t,x)\in L_2(D)\), \(b(t)\in L_2[0,T]\), then problem (A) has at most one solution almost everywhere satisfying in \(D\) the conditions

\[ |u(t,x)|<R_1,\qquad |u'_t(t,x)|<R_2,\qquad |u'_x(t,x)|<R_3. \]

§ 3. \(k\)-times \((k\geq 2)\) continuously differentiable in \(D\) solutions of problem (A).

Theorem 7. Suppose:

1. The function \(\varphi(x)\) is \(k\)-times continuously differentiable on the interval \([0,l]\), \(\varphi^{k+1}(x)\in L_2[0,l]\), \(\varphi^{(2m)}(0)=\varphi^{(2m)}(l)=0\), where \(m=0,\ldots,[k/2]\).

2. The function \(\psi(x)\) is \(k-1\) times continuously differentiable on the interval \([0,l]\), \(\psi^{(k)}(x)\in L_2[0,l]\), \(\psi^{(2m)}(0)=\psi^{(2m)}(l)=0\), where \(m=0,\ldots,[(k-1)/2]\).

3. The function \(F[\xi_1,\xi_2,\xi_3,\xi_4,\xi_5]\), defined in the domain \(Q=D\times(-R<\xi_3,\xi_4,\xi_5<R)\), has partial derivatives
   \(\partial^k F/\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4}\),

defined in the same domain, where

$$ R^2 > \frac{2 l^{2k-3}(l+\alpha \pi)^2(\max\{1,T\})^2}{45\,\alpha^2\pi^{2(k-1)}}\left[a^2\|\varphi^{(k+1)}(x)\|_{L_2[0,l]}^2+\|\psi^{(k)}(x)\|_{L_2[0,l]}^2\right]. $$

1. The functions
   \(\partial^s F/\partial \xi_1^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4}\) \((s=0,\ldots,k-2)\),
   \(\partial^s F/\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4}\) \((s=1,\ldots,k-1)\) are continuous jointly in their arguments in the domain \(Q\).

2. The functions
   \(\partial^k F/\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4}\) are measurable in \((\xi_1,\xi_2)\) for fixed \((\xi_3,\xi_4,\xi_5)\) and, for almost all \((\xi_1,\xi_2)\in D\), satisfy the condition

$$ \left| \frac{\partial^k F[\xi_1,\xi_2,\xi_3,\xi_4,\xi_5]}{\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4}} - \frac{\partial^k F[\xi_1,\xi_2,\widetilde{\xi}_3,\widetilde{\xi}_4,\widetilde{\xi}_5]}{\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4}} \right| \le a(\xi_1,\xi_2)\sum_{s=3}^{5}|\xi_s-\widetilde{\xi}_s|, $$

where \(a(\xi_1,\xi_2)\in L_2(D)\).

6.

$$ \partial^k F[\xi_1,\xi_2,0,0,0]/\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4}\in L_2(D), $$

$$ \partial^{2s}F[\xi_1,0,0,0,\xi_5]/\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4} = \partial^{2s}F[\xi_1,l,0,0,\xi_5]/\partial \xi_2^{k_1}\partial \xi_3^{k_2}\partial \xi_4^{k_3}\partial \xi_5^{k_4} \equiv 0, $$

where \(s=0,\ldots,[(k-1)/2]\).

Then, under each of the following conditions, problem (A) has a unique solution in \(D\), continuously differentiable \(k\) times, satisfying condition (2) in \(D\): a) one of the numbers \(T\), \(|\lambda|\) is fixed, and the other is sufficiently small; b) \(\lambda\), \(T\) are fixed, while
\(\|\varphi^{(k+1)}(x)\|_{L_2[0,l]}\), \(\|\psi^{(k)}(x)\|_{L_2[0,l]}\),
\(\left\|\partial^k F[\xi_1,\xi_2,0,0,0]/\partial \xi_2^k\right\|_{L_2(D)}\) are sufficiently small.

Azerbaijan State University
named after S. M. Kirov

Received
30 VII 1962

CITED LITERATURE

1. K. I. Khudaverdiev, a) Scientific Notes of Azerbaijan State University named after S. M. Kirov, Series of Physico-Mathematical and Chemical Sciences, No. 3 (1960); b) No. 4 (1960); c) No. 1 (1961); d) No. 4 (1961).