UDC 517.946

MATHEMATICS

G. I. CHANDIROV

INVESTIGATION OF A WEAK SOLUTION OF A MIXED PROBLEM FOR A HYPERBOLIC EQUATION WITH A NONLINEAR PART

(Presented by Academician A. N. Tikhonov on 2 III 1967)

The main question studied in this note is the question of the correctness and existence of a certain kind of solutions of the mixed problem for the hyperbolic equation

[
u_{tt}=\mathcal L u+F(t,x,u)
\tag{1}
]

in the cylinder (\Omega_T=[0,T]\times g),

[
u(0,x)=\varphi(x),\qquad u_t(0,x)=\psi(x);
\tag{2}
]

[
B_\Gamma u=0;
\tag{3}
]

here (g) is an (N)-dimensional domain; (\Gamma) is the boundary of this domain; (x=(x_1,x_2,\ldots,x_N)); (\varphi(x),\psi(x)) are functions prescribed in the domain (g); (F(t,x,u)) is a prescribed function defined in the strip (\Omega_T\times[-l,l]), where (0<l\leq+\infty). (\mathcal L) is a self-adjoint differential expression which, together with condition (3), generates a self-adjoint positive operator; denote by (D_L) the domain of definition of this operator.

Mixed problems for quasilinear equations have been studied by L. Lichtenstein, M. R. Siddiqi, D. Levi, M. Krzyżański, I. Schauder, W. Barbăuti, A. I. Guseinov, K. I. Khudaverdiev, K. K. Gasanov, V. N. Gol’dberg, Yu. N. Neimark, S. A. Zhautykov, A. Sh. Guseinov, G. N. Khalilov, and others. In contrast to the works of these authors, we study a weak (in a certain sense) solution of problem (1), (2), (3).

It is clear that the operator (\mathcal L) is invertible. Let some iteration of the inverse operator (\mathcal L^{-1}) be a completely continuous operator in (\mathcal L_2(g)). Obviously, (\mathcal L) has a complete orthonormal system of eigenfunctions ({v_n(x)}). Denote by ({\lambda_n^2}) the system of eigenvalues corresponding to ({v_n(x)}); by (W_2(\mathcal L)) the set of functions (u(t,x)) which, for any (t\in[0,T]), are elements of (D_{\mathcal L}) for all (x\in g) and have continuous second derivatives with respect to (t), with (u(t,x)\equiv0) if (t\in(T-\delta,T)), where (0<\delta<T); by (\mathcal L_2(\Omega_T)) the set of measurable functions (a(t,x)) for which

[
\int_0^T\int_g a^2(t,x)\,dx\,dt<+\infty;
]

and by (B_\alpha) (where (\alpha=(a,2,m))) the set of functions from (\mathcal L_2(\Omega_T)) for which the Fourier coefficients with respect to the eigenfunctions of the operator (\mathcal L) satisfy the inequality

[
\sum_{n=1}^{\infty}\left[\sum_{k=0}^{m}\lambda_n^{\alpha_1}
\left(\max_{0\leq t\leq T}\left|A_n^{(k)}(t)\right|\right)\right]^2<+\infty.
]

A function (F(t,x,u)) of ((N+2)) variables, (t\in[0,T]), (x\in g), (-l\leq u\leq l), is said to satisfy the Carathéodory conditions if (F) is continuous in (u) for almost all ((t,x)\in\Omega_T) and measurable in ((t,x)) for every (u\in[-l,l]). We shall assume that (F(t,x,u)) satisfies the Carathéodory conditions.

If (u(t,x)\in \mathcal L_2(\Omega_T)) satisfies the integral identity

[
\int_0^T\int_g {u[\Phi_{tt}-\mathcal L\Phi]-F(t,x,u)\Phi(t,x)}\,dx\,dt
+\int_g\left[\psi(x)\frac{\partial\Phi}{\partial t}-\varphi(x)\Phi\right]_{t=0}dx=0
]

for every (\Phi(t,x)\in W_2(\mathcal L)), then the function (u(t,x)), following S. L. Sobolev, will be called an ((\mathcal L_2,W_2(\mathcal L)))-weak solution of problem (1), (2), (3) (briefly, an ((\mathcal L_2,W_2(\mathcal L)))-w.s.).

In studying the correctness and existence of an ((\mathcal L_2,W_2(\mathcal L)))-w.s. we use an inequality of Gronwall type, contained in Lemma 1, and the continuity of the operator (F).

Lemma 1. Let (E) be a partially ordered Banach space, (F) a nondecreasing continuous operator in (E), and (u\leq F(u)), (v_{n+1}=F(v_n)), where (v_0=u). Then, if (v_n\to v) as (n\to+\infty), then (u\leq v).

From this lemma there follows the following assertion. If a nonnegative function (a(t)\in C_{(0,T)}) satisfies the inequality

[
a(t)\leq C+\int_0^t [b(\tau)a(\tau)+m(\tau)a^\alpha(\tau)]\,d\tau,
]

where (C\geq0) is a constant, (\alpha\in[0,1]), and the functions (b(t)), (m(t)) are nonnegative and integrable on ((0,T)), then

[
a(t)\leq\left{C^{1-\alpha}\exp\left[(1-\alpha)\int_0^t m(\tau)\,d\tau\right]+\right.
]

[
\left.
+(1-\alpha)\int_0^t b(\tau)\exp\left[(1-\alpha)\int_\tau^t m(r)\,dr\right]\,d\tau\right}^{1/(1-\alpha)}.
]

Theorem 1. Let the Carathéodory condition hold and let (l=+\infty). Then, if

[
|F(t,x,u)|\leq a(t,x)+b(x)|u|,
]

where (a(t,x)\in \mathcal L_2(\Omega_T)), (b(t)\in \mathcal L_2(0,T)), then (F) acts from (B_{(0,2,0)}(T)) into (\mathcal L_2(\Omega_T)), is continuous and bounded.

Lemma 2. If: a) (F(t,x,u)) satisfies the conditions of Theorem 1; b) (u(t,x)) is an ((\mathcal L_2,W_2(\mathcal L)))-w.s.; c)

[
A_n(t)=\int_g u(t,x)v_n(x)\,dx,\qquad n=1,2,\ldots,
]

then the Fourier coefficients (A_n(t)) satisfy the countable system of nonlinear integral equations

[
A_n(t)=\varphi_n\cos\lambda_n t+\frac{\psi_n\sin\lambda_n t}{\lambda_n}+
]

[
+\frac{1}{\lambda_n}\int_0^t\int_g F\left[\tau,x,\sum_{k=1}^{\infty}A_k(\tau)v_k(x)\right]v_n(x)\sin\lambda_n(t-\tau)\,dx\,d\tau,
\tag{A}
]

where

[
\varphi_n=\int_g \varphi(x)v_n(x)\,dx,\qquad
\psi_n=\int_g \psi(x)v_n(x)\,dx,\qquad n=1,2,\ldots .
]

Using system (A), one can prove the following theorems.

Theorem 2. Let: a) (\varphi_i(x)), (\psi_i(x)), (i=1,2), be elements of the space (\mathcal L_2(g)); b) (F_i(t,x,0)\in \mathcal L_2(\Omega_T)), and for any (u,v) from ((-\infty,\infty)) the inequality

[
|F_1(t,x,u)-F_2(t,x,v)|\leq k(t)|u-v|+C(t,x),
]

holds, where (C(t,x)\in \mathcal L_2(\Omega_T)), (k(t)\geq0) and is integrable on ((0,T)); c) (u_i(t,x)) are ((\mathcal L_2,W_2(\mathcal L)))-w.s., i.e.

[
\int_0^T\int_g {u_i[\Phi_{tt}-\mathcal L\Phi]-F_i(t,x,u_i)\Phi}\,dx\,dt
+\int_g\left[\psi_i\frac{\partial\Phi}{\partial t}-\varphi_i\Phi\right]_{t=0}dx=0,
]

[
i=1,2.
\tag{4}
]

Then, if (A_n^{(i)}(t)=\displaystyle\int_g u_i(t,x)u_n(x)\,dx,\ i=1,2;\ n=1,2,\ldots,) the inequality

[
\sum_{n=1}^{\infty}\max_{0\le t\le T}\left|A_n^{(2)}(t)-A_n^{(1)}(t)\right|
\le
3\left{
\int_g [\varphi_2(x)-\varphi_1(x)]^2\,dx
+\frac{1}{\lambda_1}\int_g [\psi_2(x)-\psi_1(x)]^2\,dx
+\int_0^T\int_g C^2(t,x)\,dx\,dt
\right}
]
[
+\int_0^T
\left{
k(s)\left[\int_0^s\int_g C^2(t,x)\,dx\,dt\right]
\left(\exp\int_s^T k(r)\,dr\right)
\right}\,ds .
]

Theorem 3. Suppose: a) the sequence of eigenfunctions of the operator (\mathcal L) is uniformly bounded, i.e. there exists a number (M) such that (|v_n(x)|\le M) for every (n); b) (u_i(t,x)) satisfy the integral identity (4), i.e. are ((\mathcal L_2,W_2(\mathcal L)))-w.s.; c) in the Carathéodory condition (l=+\infty),

[
\int_0^T
\left[
\int_g |F_i(t,x,u_i(t,x))|^2\,dx
\right]^{q/p}dt<+\infty,\qquad i=1,2,
]

where (1/p+1/q=1,\ p>1) is such that

[
\sum_{n=1}^{\infty}\frac{1}{\lambda_n^p}<+\infty;
]

d)

[
|F_2(t,x,u_2(t,x))-F_1(t,x,u_1(t,x))|
\le
a_0(t,x)|u_2-u_1|+a_1(t,x)|u_2-u_1|^\alpha+a_2(t,x),
]

where (0\le \alpha\le 1),

[
\int_0^T
\left[
\int_g a_i^p(t,x)\,dx
\right]^{q/p}dt<+\infty,\qquad i=0,1,2.
]

Then, if

[
A_n^{(i)}(t)=\int_g u_i(t,x)v_n(x)\,dx,\qquad
b_n^{(i)}(t)=\varphi_n^{(i)}\cos\lambda_n t+\psi_n^{(i)}\frac{\sin\lambda_n t}{\lambda_n},
]

where

[
\varphi_n^{(i)}=\int_g \varphi_i(x)v_n(x)\,dx,\qquad
\psi_n^{(i)}=\int_g \psi_i(x)v_n(x)\,dx,\qquad i=1,2;\ n=1,2,\ldots,
]

the inequalities

[
\sum_{n=1}^{\infty}\max_{0\le t\le T}\left|A_n^{(2)}(t)-A_n^{(1)}(t)\right|
\le
\left{
\left[
\alpha_0\left(
\sum_{n=1}^{\infty}\max_{0\le t\le T}|b_n^{(2)}(t)-b_n^{(1)}(t)|
+\alpha_1\int_0^T
\left(\int_g a_2^p(t,x)\,dx\right)^{q/p}dt
\right)
\right]^{1-\alpha}
\right.
]
[
\left.
\cdot
\exp(1-\alpha)\,\alpha_2\int_0^T
\left(\int_g a_1^p(t,x)\,dx\right)^{q/p}dt
+
(1-\alpha)\int_0^T
\left(\int_0^\tau\int_g a_2(\xi,x)\,dx\,d\xi\right)
\exp\int_\tau^T
\left[
\int_g a_0^p(\eta,x)\,dx
\right]^{q/p}d\eta\,d\tau
\right}^{1/q(1-\alpha)},
]

where (\alpha_0,\alpha_2,\alpha_1) are constants.

Theorem 4. Suppose: a) in the Carathéodory condition (l=+\infty); b) (F(t,x,u(t,x))\in \mathcal L_2(\Omega_T)), if (u(t,x)) is an ((\mathcal L_2,W_2(\mathcal L)))-w.s.; c)

[
|F(t,x,u)-F(t,x,v)|\le \frac{|u-v|}{t},\qquad \lambda_1\ge 1.
]

Then problem (1), (2), (3) cannot have more than one ((\mathcal L_2,W_2(\mathcal L)))-w.s.

It is not difficult to show that if: a) the boundary of the domain (g) is such that Green’s formula is applicable; b) (\varphi(x),\psi(x)) are elements of (\mathcal L_2(g)); c) (|F(t,x,u)|\le a(t,x)+b|u|) for all (u\in(-\infty,\infty)), where (a(t,x)\in \mathcal L_2(\Omega_T)), (b) is a constant, and the sequence ({A_n(t)}) is a solution of system (A), then the function

[
u(t,x)=\sum_{n=1}^{\infty} A_n(t)v_n(x)
]

is an ((\mathcal L_2,W_2(\mathcal L)))-w.s. (here (v_n(x)) are eigenfunctions of the operator (\mathcal L)).

Theorem 5. Suppose: a) in the Carathéodory condition (l=+\infty); b) (\varphi(x)), (\psi(x)\in \mathscr L_2(g)); c)

[
|F(t,x,u)|\le a(t,x)+a_1(t,x)|u|^\alpha+b(t)|u|;
]

here (a(t,x)\in \mathscr L_2(\Omega_T)), (b(t)\in \mathscr L_2(0,T)), (b(t)>0),

[
\int_0^T\left[\int_g a_1^{\,2/(1-\alpha)}(t,x)\,dx\right]^{1-\alpha}dt<+\infty .
]

Then problem (1), (2), (3) has at least one ((\mathscr L_2,W_2(\mathscr L)))-s.r., and there exist some positive numbers (T,R) such that

[
\sum_{n=1}^{\infty}\max_{0\le t\le T}|A_n(t)|^2\le R^2,
]

where (A_n(t)) are the Fourier coefficients of the ((\mathscr L_2,W_2(\mathscr L)))-s.r. with respect to the eigenfunctions of the operator (\mathscr L).

Theorem 6. Suppose: a) the set of eigenvalues of the operator (\mathscr L) is bounded by some number (M), i.e. (|v_n(x)|\le M) for any (n);

b) (|F(t,x,u)|\le a(t,x)) for all (u\in(-a,a)), where

[
\int_0^T\left[\int_g a^p(t,x)\,dx\right]^{q/p}dt<+\infty
]

and the number (p) is such that

[
\sum_{n=1}^{\infty}\frac1{\lambda_n^p}<+\infty,
]

(q) is conjugate to (p);

c) (\varphi(x)), (\psi(x)), (T) are such that

[
\sum_{p=1}^{\infty}\left(|\varphi_n|+\frac1{\lambda_n}|\psi_n|\right)
\left(\sum_{n=1}^{\infty}\frac1{\lambda_n^p}\right)^{1/p}
M_1\left{\int_0^T\left[\int_g a^p(t,x)\,dx\right]^{1/p}dt\right}^{1/q}