MATHEMATICS

A. Kh. GELIG

ON THE STABILITY OF SOLUTIONS OF THE CAUCHY PROBLEM AND THE MIXED PROBLEM FOR HYPERBOLIC EQUATIONS

(Presented by Academician V. I. Smirnov on 2 VI 1958)

Let \(\varphi\) be a function of \(t, x_1,\ldots,x_n\). By \(D^p\varphi\) we shall denote any generalized derivative of \(\varphi\) with respect to \(t, x_1,\ldots,x_n\) of order \(p\), and by the index \((p)\) the summation over all such derivatives. In what follows, summation over the indices \(i,j,s\) from \(1\) to \(n\) is assumed. We introduce the notation:
\[ \|\varphi\|^2_{W^{(k)}_2(E_n)} = \sum_{p=0}^{k}\sum_{(p)}\int_{E_n}|D^p\varphi|^2\,dx, \qquad E(\varphi)\equiv a_{ij}\frac{\partial\varphi}{\partial x_i}\frac{\partial\varphi}{\partial x_j} + \left(\frac{\partial\varphi}{\partial t}\right)^2 + \beta\varphi^2, \]
\[ \|\varphi\|^q_{L_q}\equiv \sup \int |\varphi|^q\,dx, \]
where the integration is carried out over a ball of unit radius, and the supremum is taken over all such balls.

1. Consider the Cauchy problem for the equation
   \[ u_{tt} = a_{ij}u_{x_i x_j} + 2a_{i0}u_{x_i t} + a_i u_{x_i} + (a_0-\alpha)u_t + (a-\beta)u + f, \tag{1} \]
   \[ u|_{t=0}=\varphi_0(x_1,\ldots,x_n), \qquad u_t|_{t=0}=\varphi_1(x_1,\ldots,x_n). \tag{2} \]
   The coefficients and \(f\) depend on \(t,x_1,\ldots,x_n\) and in Theorem 1, together with the functions (2), satisfy the conditions of the existence and uniqueness theorem for a generalized solution in the sense of S. L. Sobolev \((^1)\), while in Theorem 2 they possess those generalized derivatives which occur in the statement of the theorem. In addition, \(a_{lm}, D^1a_{lm}, a_l\) are assumed continuous.

Suppose that for \(t\ge 0\) the inequality
\[ a_{ij}\xi_i\xi_j\ge \tau \xi_i^2,\qquad \tau=\mathrm{const}>0 \tag{3} \]
holds.

Denote by \(M\) and \(L\) the maxima of the coefficients \(|a_{lm}|\) and \(|a_{l0}|\) \((1\le m,l\le n)\) in the cylinder
\[ C\,[0\le t\le T;\; 0\le x_1^2+\cdots+x_n^2\le r^2], \]
and suppose that for any \(d>0\) and \(T>0\)
\[ \int_d^\infty \frac{dr}{L+\sqrt{L^2+M}}=\infty. \tag{4} \]

Theorem 1. Suppose that the coefficients of equation (1) satisfy the following conditions:
\[ 1)\quad \frac{\partial a_{ij}}{\partial t}\xi_i\xi_j\le 0 \quad \text{for } t>0 \quad \text{or}\quad |a_{lm}|\le A(t)\;(1\le l,m\le n); \]
\[ 2)\quad \sup_{-\infty<x_1,\ldots,x_n<\infty} \left| a_m-\sum_{l=1}^{n}\frac{\partial a_{lm}}{\partial x_l} \right| \le A(t)\;(0\le m\le n); \qquad \int_0^\infty A(t)\,dt<\infty; \]
\[ 3)\quad \|a\|_{L_q}\le A_q(t), \qquad q=n\;(n>2),\quad 2+\gamma\;(n=2),\quad 2\;(n=1); \]
here and below \(\gamma\) is an arbitrarily small positive number, and the function \(A_q(t)\) is bounded and summable from \(0\) to \(\infty\);
\[ 4)\quad \alpha(t,x_1,\ldots,x_n)\ge 0,\qquad \beta(x_1,\ldots,x_n)\ge \beta_0=\mathrm{const}>0. \]

Then \(\|u\|_{W_2^{(1)}(E_n)}<\varepsilon\) for all \(t>0\), if

\[ \int_{E_n}\left[\left.a_{ij}\right|_{t=0}\frac{\partial\varphi_0}{\partial x_i}\frac{\partial\varphi_0}{\partial x_j} +\varphi_1^2+\beta\varphi_0^2\right]\,dx +\int_0^\infty \|f\|_{L_2(E_n)}\,dt<\delta(\varepsilon). \]

Theorem 2. Suppose that the coefficients of the equation satisfy, for \(k>1\), the following requirements:

\[ \begin{gathered} 1)\quad \|D^1 a_{ml}\|_C,\ \|D^p a_{ml}\|_{L_{q(p)}}\le A(t) \quad (1\le m\le n,\ 0\le l\le n),\\ \text{where } q(p)=\frac{n}{p-1}\left(2\le p<\left[\frac n2\right]\right),\quad \frac{2n}{n-1}\left(p=\left[\frac n2\right]+1\right),\\ 2\left(\left[\frac n2\right]+1<p\le k-1\right); \\[0.5em] 2)\quad \|a_m\|_C,\ \|D^p a_m\|_{L_{s(p)}}\le A(t)\quad (0\le m\le n),\ \text{where }\\ s(p)=\frac np\left(1\le p<\left[\frac n2\right]\right),\quad \frac{2n}{n-1}\left(p=\left[\frac n2\right]\right),\quad 2\left(\left[\frac n2\right]<p\le k-1\right); \\[0.5em] 3)\quad \|a\|_{q}\le A_q(t);\quad q,\ A_q(t),\ A(t)\text{ are the same as in Theorem 1;}\\ \|D^p a\|_{L_{\sigma(p)}}\le A(t),\ \text{where } \sigma(p)=\frac{n}{p+1}\left(1\le p<\left[\frac n2\right]-1\right),\quad \frac{2n}{n-1}\left(p=\left[\frac n2\right]-1\right),\\ 2\left(\left[\frac n2\right]-1<p\le k-1\right); \\[0.5em] 4)\quad \alpha=\mathrm{const}\ge 0,\quad \beta=\mathrm{const}>0. \end{gathered} \]

Then \(\|u\|_{W_2^{(k)}(E_n)}<\varepsilon\) for all \(t>0\), if

\[ \sum_{p=0}^{k-1}\sum_{(p)} \left[ \int_{E_n}\left.E(D^p u)\right|_{t=0}\,dx +\int_0^\infty \|D^p f\|_{L_2(E_n)}\,dt \right]<\delta(\varepsilon). \]

If \(\alpha>0\), then, with an insignificant change in the conditions of Theorems 1 and 2, one can prove the asymptotic stability of the solution \(u\equiv 0\). Examples show that condition 4) in Theorems 1 and 2 cannot be improved.

Consider the quasilinear equation

\[ u_{tt}=a_{ij}u_{x_i x_j}+2a_{i0}u_{x_i t}-\alpha u_t-\beta u+f+\bar f \tag{5} \]

with the initial condition (2). Here \(a_{ij}, a_{i0}, f\), and \(\bar f\) are continuous functions of
\(t, x_1,\ldots,x_n, u, u_t, u_{x_1},\ldots,u_{x_n}\), defined in the domain
\(\mathfrak D_1=[t\ge 0,\ -\infty<x_1,\ldots,x_n<+\infty]\times\mathfrak D_2\),
\(\mathfrak D_2=[|u|,\ |u_t|,\ |u_{x_1}|,\ldots,|u_{x_n}|<\Delta]\),
having continuous derivatives with respect to \(u, u_t, u_{x_1},\ldots,u_{x_n}\) up to some order, which in turn have generalized partial derivatives with respect to
\(t, x_1,\ldots,x_n\). The order of the derivatives will be indicated in Theorem 3. Suppose that
\(\dot f(t,x_1,\ldots,x_n,0,\ldots,0)\equiv 0\). Differentiate \(\mu\) times \(a_{ml}\) with respect to
\(u, u_t, u_{x_1},\ldots,u_{x_n}\) and \(\nu\) times with respect to the remaining arguments, and substitute into the resulting derivative, in place of \(u\), any continuously differentiable function from \(\mathfrak D_2\). The composite function of \(t,x_1,\ldots,x_n\) obtained in this way will be denoted by \(D_\mu^\nu a_{ml}\). Let \(a_{ml}\), after substituting into them any function from \(\mathfrak D_2\), satisfy, for \(t\ge 0\), conditions (3) and (4).

Theorem 3. Suppose that, for \(\eta=\max([n/2]+2,3)\), the following inequalities hold:

\[ \begin{gathered} 1)\quad \|D^1 a_{ml}\|_C,\ \|D_1 a_{ml}\|_C,\ \|D_\mu^\nu a_{ml}\|_{L_{q(\nu+\mu)}}\le A_q(t),\\ 2\le \nu+\mu\le \eta,\quad 1\le m\le n,\ 0\le l\le n; \\ 2)\quad \|D_1^1 f\|_C,\ \|D_1 f\|_C,\ \|D_{\mu+1}^{\nu} f\|_{L_{q(\nu+\mu)}}\le A_q(t),\quad 2\le \nu+\mu\le \eta; \\ 3)\quad \|D_1\bar f\|_C,\ \|D_\mu^\nu \bar f\|_{L_{q(\nu+\mu)}}\le A_q(t),\quad 2\le \nu+\mu\le \eta,\quad \mu>0;\quad \|D^{\eta+1}\bar f\|_{L_2},\\ \|D_1^\eta\bar f\|_{L_2}\le \mathrm{const};\quad \text{here } q(p)\text{ is the same as in condition 1) of Theorem 2;}\\ A_q\text{ are continuous functions of }t,\text{ integrable from }0\text{ to }\infty; \\ 4)\quad \alpha=\mathrm{const}\ge 0,\quad \beta=\mathrm{const}>0. \end{gathered} \]

Then, if

\[ \sum_{p=0}^{\eta}\sum_{(p)}\left[\int_{E_n} E\,(D^p u)\big|_{t=0}\,dx+\int_0^\infty \|D^p\bar f\|_{L_2(E_n)}\,dt\right]<\delta(\varepsilon), \tag{6} \]

there exists, for all \(t>0\), a classical solution of equation (5), and
\(\|u\|_{W_2^{(\eta+1)}(E_n)},\ |u|_{C^s}<\varepsilon\).

1. Let a system be given

\[ A^0(t,x)U_t=A^s(t,x)U_{x_s}+B(t,x)U+F(t,x) \tag{7} \]

with the initial condition

\[ U(0,x)=\Phi(x). \tag{8} \]

Here \(A^\rho\) are Hermitian matrices with elements \(a_{ml}^{\rho}\); \(B\) is a square matrix with elements \(b_{ml}\); \(U,F\), and \(\Phi\) are columns with elements \(u_m, f_m, \varphi_m\) \((0\le \rho\le n,\ 1\le m,l\le N)\).

Suppose that for \(t\ge 0\) the condition

\[ (A^0\Xi,\Xi)\ge \tau_2(\Xi,\Xi),\qquad \tau_2=\mathrm{const}>0 \tag{9} \]

is satisfied.

Denote by \(M\) the maximum of \(|a_{ml}^{\rho}|\) \((1\le \rho\le n)\) in the cylinder \(\Pi\), and suppose that for all \(d>0,\ T>0\)

\[ \int_d^\infty \frac{dr}{M}=\infty. \tag{10} \]

Assume that in any bounded part of the half-space \(t\ge 0\),
\(a_{ml}^{\rho}\in C^1,\ b_{ml}\in C,\ \|D^1 b_{ml}\|_{L_q}\le \mathrm{const}\)
\((q=n\ (n>2),\ 2+\gamma\ (n=2),\ 2\ (n=1))\),
\(\varphi_m\in W_2^{(1)},\ f_m\in W_2^{(1)}\). Then the following holds:

Theorem 4. Suppose that
\[ A_t^0-A_{x_s}^s+B+B^*=C+\Gamma, \]
where \(C\) is a constant nonnegative Hermitian matrix, and
\[ |\Gamma(t,x)|\le A(t). \]
Here \(b_{ml}^*=\overline{b}_{lm}\), and the function \(A(t)\) is summable from \(0\) to \(\infty\).

Then
\[ \|u_m\|_{L_2(E_n)}<\varepsilon\quad (1\le m\le N) \]
for all \(t>0\), if

\[ \int_{E_n}(A^0|_{t=0}\Phi,\Phi)\,dx +\sum_{m=1}^{N}\int_0^\infty \|f_m\|_{L_2(E_n)}\,dt <\delta(\varepsilon). \]

Theorem 5. Let the coefficients \(a_{ml}^{\rho}\) \((0\le \rho\le n,\ 1\le l,m\le N)\) satisfy condition 1) of Theorem 2 for \(k>1\); \(B=C+G\), where \(C\) is a constant nonnegative Hermitian matrix, and the elements \(g_{ml}(t,x)\) of the matrix \(G\) satisfy condition 2) of Theorem 2.

Then
\[ \|u_m\|_{W_2^{(k-1)}(E_n)}<\varepsilon \]
for \(t>0\), if

\[ \sum_{p=0}^{k-1}\sum_{(p)} \left[ \int_{E_n}(A^0D^pU,D^pU)\big|_{t=0}\,dx +\int_0^\infty \sum_{m=1}^{N}\|D^p f_m\|_{L_2(E_n)}\,dt \right]<\delta(\varepsilon). \]

Consider, with the initial condition (8), the system

\[ A^0U_t=A^sU_{x_s}+BU+F(t,x,U)+\bar F(t,x,U),\qquad F(t,x,0)\equiv 0 \tag{11} \]

with real symmetric matrices \(A^\rho(t,x,U)\) \((0\le \rho\le n)\) and a real constant matrix \(B\), where in the domain
\(\mathfrak D_3=[t\ge 0,\ -\infty<x_1,\ldots,x_n<+\infty]\times \mathfrak D_4\),
\(\mathfrak D_4=[|u_1|,\ldots,|u_N|<\Delta]\), the requirements (9) and (10) are fulfilled.

Theorem 6. If \(a_{ml}^{p}\) satisfy condition 1), \(f_m\) condition 2), \(\bar f_m\) condition 3) of Theorem 3 \((0 \leqslant p \leqslant n,\ 1 \leqslant m,\ l \leqslant N)\), \(B+B' = C\) \((b_{ml}=b_{lm})\), where \(C\) is a nonnegative matrix, and

\[ \sum_{p=0}^{n}\sum_{(p)}\left[ \int_{E_n}(A^0D^pU,D^pU)\big|_{t=0}\,dx + \int_0^\infty \sum_{m=1}^{N}\|D^p\bar f_m\|_{L_2(E_n)}\,dt \right] < \delta(\varepsilon), \]

then for all \(t>0\) there exists a classical solution of equation (11), and
\[ \|u_m\|_{W_2^{(n)}(E_n)},\ \|u_m\|_{C^1} < \varepsilon \quad (1 \leqslant m \leqslant N). \]

1. Let, in the rectangular parallelepiped
   \(Q[0 \leqslant t \leqslant T,\ 0 \leqslant x_1,\ldots,x_n \leqslant K]\), the coefficients of the equation

\[ u_{tt}=a_{ij}u_{x_i x_j}+2a_{i0}u_{x_i t}+a_i u_{x_i}+a_0u+(a-\beta)u+f,\quad \beta=\mathrm{const}, \tag{12} \]

and their derivatives have the bounded norms specified in conditions 1)—3) of Theorem 2, \(f\in W_2^{(k-1)}(Q)\), and inequality (3) is satisfied. Denote by \(F\) the face of the parallelepiped lying in the hyperplane \(x_n=0\), by \(\Omega\) the face in the hyperplane \(t=0\), and by \(\Gamma\) the intersection of \(F\) with \(\Omega\). On \(\Omega\) the initial conditions (2) are prescribed, and on \(F\) the boundary condition

\[ u\big|_{x_n=0}=\psi(t,x_1,\ldots,x_{n-1}). \tag{13} \]

Consider a truncated pyramid \(R\), lying inside \(Q\) and bounded by the hyperplanes \(t=0,\ t=T,\ x_n=0\) and by an oriented space-like characteristic surface \(S\), whose exterior normal makes an acute angle with the positive direction of the \(t\)-axis.

Lemma. If \(\varphi_0\in W_2^{(k)}(\Omega)\), \(\varphi_1\in W_2^{(k-1)}(\Omega)\), \(\psi\in W_2^{(k)}(F)\), and on \(\Gamma\) the compatibility conditions up to order \(k-1\) inclusive are satisfied, then in \(R\) there exists a solution of equation (12), \(u\in W_2^{(k)}(R)\). Derivatives of order \(k\) have limiting values on the section \(\Omega_t\) of the pyramid \(R\) by the hyperplane \(t=\mathrm{const}\) in the sense that there exist sequences of continuous functions converging to \(D^p u\) weakly in \(L_2(R)\), which also converge weakly in \(L_2(\Omega_t)\).

In this lemma, the number of required generalized derivatives of \(\varphi_0,\varphi,\psi\) and the order of the compatibility conditions are lowered by one in comparison with the result of O. A. Ladyzhenskaya \((^2)\).

Consider the mixed problem (2), (13) for equation (12) in the domain
\[ V=[t\geqslant 0,\ x_n\geqslant 0,\ -\infty<x_1,\ldots,x_{n-1}<+\infty]. \]
Let \(\varphi_0\in W_2^{(k)}(\Omega)\), \(\varphi_1\in W_2^{(k-1)}(\Omega)\), \(\psi\in W_2^{(k)}(F)\), \(f\in W_2^{(k-1)}(V)\), where
\(\Omega=[x_n\geqslant 0,\ t=0]\), \(F=[x_n=0,\ t\geqslant 0]\), and on
\(\Gamma[t=0,\ x_n=0]\) the compatibility conditions up to order \(k-1\) are satisfied. Suppose also that inequality (3) holds in \(V\). Then the following is true:

Theorem 7. If in the domain \(V\), for \(k>1\), conditions 1)—3) of Theorem 2 are satisfied and, moreover,
\[ |a_{ml}|\leqslant \mathrm{const}\quad (1\leqslant m\leqslant n;\ 0\leqslant l\leqslant n),\quad A(t),\ A_q(t)\leqslant \mathrm{const}, \]
and \(\beta\) is a positive constant, then
\[ \|u\|_{W_2^{(k)}(E_n)}<\varepsilon \]
for all \(t>0\), if

\[ \|u|_{t=0}\|_{W_2^{(k)}(\Omega)}^{2} + \sup_{0<t<\infty} \sum_{p=0}^{k-2}\sum_{(p)}\|D^p f\|_{L_2(\Omega)}^{2} + \|\psi\|_{W_2^{(k)}(F)}^{2} + \]

\[ + \int_0^\infty \sum_{p=0}^{k-1}\sum_{(p)} \left( \|D^p f\|_{L_2(\Omega)} + \|D^p f\|_{L_2(\Omega)}^{2} \right)\,dt <\delta(\varepsilon). \]

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
28 V 1958

REFERENCES

1. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
2. O. A. Ladyzhenskaya, The mixed problem for a hyperbolic equation, 1953.