L. Ya. Plyanzina

ON THE APPROXIMATION OF SOLUTIONS OF BOUNDARY-VALUE PROBLEMS FOR PARABOLIC AND HYPERBOLIC EQUATIONS BY MEANS OF SOLUTIONS OF THE CAUCHY PROBLEM

(Presented by Academician L. S. Pontryagin on 15 XI 1963)

In the present paper, solutions of boundary-value problems for parabolic and hyperbolic equations of second order are obtained as limits, as \(\varepsilon \to 0\), of solutions of the Cauchy problem for certain equations with coefficients depending on \(\varepsilon\). Similar problems were considered in works \((^{1-3})\), etc.

1. In the cylinder \(Q^{-}=\{\Omega^{-}\times[0T]\}\), where \(\Omega^{-}\) is a bounded domain of the space \(E_n\) with boundary \(\Gamma\), a parabolic equation is given

\[ a(x,t)\frac{\partial u}{\partial t} + \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( b_{ij}(x,t)\frac{\partial u}{\partial x_j} \right) + \sum_{i=1}^{n} c_i(x,t)\frac{\partial u}{\partial x_i} + d(x,t)u = \varphi(x,t). \tag{1} \]

Let \(a,b_{ij},c_i,d,\varphi,\partial a/\partial t,\partial b_{ij}/\partial t,\partial d/\partial t\) \((i,j=1,2,\ldots,n)\) be bounded measurable functions, \(d(x,t)\leqslant 0\),

\[ a(x,t)\leqslant \bar a<0,\qquad \bar\lambda_1\sum_{i=1}^{n}\xi_i^2 \geqslant \sum_{i,j=1}^{n} b_{ij}(x,t)\xi_i\xi_j \geqslant \lambda_1\sum_{i=1}^{n}\xi_i^2 \tag{2} \]

for any real vector \(\xi=(\xi_1,\xi_2,\ldots,\xi_n)\), \(\bar\lambda_1\geqslant\lambda_1>0\). The boundary of the cylinder \(S=\{\Gamma\times[0T]\}\) belongs to the class \(A_2^2\) (see \((^4)\)). We introduce, as usual, the functional spaces \(W_2^{1,1}(Q^{-})\), \(\overset{0}{W}{}_{2}^{1,1}(Q^{-})\), \(W_2^1(\Omega^{-})\), \(\overset{0}{W}{}_{2}^{1}(\Omega^{-})\) (see \((^{5,6})\), p. 93).

For equation (1) the following problems in \(Q^{-}\) are considered:

1) the first boundary-value problem with conditions

\[ u(x,t)\big|_{t=0}=f(x),\qquad f(x)\in \overset{0}{W}{}_{2}^{1}(\Omega^{-}); \tag{3} \]

\[ u(x,t)\big|_{S}=0; \tag{4} \]

2) the second boundary-value problem with condition (3) and

\[ \frac{\partial u}{\partial \nu}\bigg|_{S}=0,\qquad \text{where }\ \frac{\partial u}{\partial \nu} = \sum_{i,j=1}^{n} b_{ij}\frac{\partial u}{\partial x_j}\cos(nx_i); \tag{5} \]

3) the third boundary-value problem with condition (3) and

\[ \frac{\partial u}{\partial \nu}+Fu\bigg|_{S}=0, \tag{6} \]

where \(n\) is the outward normal to \(S\); \(F(x,t)>0\), \(F\subset C^1\) in \(\{E_n\times[0T]\}\).

In the case of problems (1), (3), (5) it is also assumed that the functions \(\partial a/\partial x_k\), \(\partial b_{ij}/\partial x_k\), \(\partial c_i/\partial x_k\), \(\partial d/\partial x_k\), \(\partial\varphi/\partial x_k\) \((i,j,k=1,2,\ldots,n)\) are measurable and bounded.

By a generalized solution of problem (1), (3), (4) we shall mean a function \(u(x,t)\in \overset{0}{W}{}_{2}^{1,1}(Q^{-})\) such that \(u(x,0)=f(x)\) and, for any function \(\Phi(x,t)\in \overset{0}{W}{}_{2}^{1,1}(Q^{-})\), \(\Phi(x,T)=0\), the integral identity

\[ \Psi(u,\Phi) \equiv \int\!\!\!\int_{Q^{-}} \left[ a\frac{\partial u}{\partial t}\Phi - \sum_{i,j=1}^{n} b_{ij} \frac{\partial u}{\partial x_j} \frac{\partial \Phi}{\partial x_i} + \sum_{i=1}^{n} c_i \frac{\partial u}{\partial x_i}\Phi + du\Phi - \varphi\Phi \right]\,dx\,dt =0. \tag{7} \]

By a generalized solution of problem (1), (3), (5) we shall mean a function \(u(x,t)\in W^{1,1}_2(Q^-)\) such that \(u(x,0)=f(x)\) and, for any function \(\Phi(x,t)\in W^{1,1}_2(Q^-)\), \(\Phi(x,T)=0\), the integral identity (7) is satisfied.

By a generalized solution of problem (1), (3), (6) we shall mean a function \(u(x,t)\in W^{1,1}_2(Q^-)\) such that \(u(x,0)=f(x)\) and, for any function \(\Phi(x,t)\in W^{1,1}_2(Q^-)\), \(\Phi(x,T)=0\), the integral identity is satisfied:
\[ \Psi(u,\Phi)-\int_S Fu\Phi\,ds=0. \tag{8} \]

Let us extend the coefficients of equation (1) to \(Q^+=\{\Omega^+\times[0T]\}\), where \(\Omega^+=E_n/\overline{\Omega^-}\), as functions \((t,x,\varepsilon)\) in such a way that, in the case of problem (1), (3), (5),
\[ |a|\le K\varepsilon^\alpha,\quad |\partial a/\partial t|\le K\varepsilon^\alpha,\quad |b_{ij}|\le K\varepsilon^\beta,\quad |\partial b_{ij}/\partial t|\le K\varepsilon^\beta, \tag{9} \]
\[ |c_i|\le K\varepsilon^\gamma,\quad |d|\le K\varepsilon^\delta,\quad |\partial d/\partial t|\le K\varepsilon^\delta,\quad |\varphi|\le K\varepsilon^\eta \quad (i,j=1,2,\ldots,n), \]
where \(\varphi(x,t,\varepsilon)=0\) outside a certain finite domain, \(d(x,t,\varepsilon)\le 0\),
\[ \eta\ge \alpha/2,\quad \gamma\ge(\beta+\alpha)/2,\quad \alpha\le\beta, \tag{10} \]
the constant \(K\) does not depend on \(\varepsilon\),
\[ a(x,t,\varepsilon)\le \tilde a(\varepsilon)<0,\quad \varepsilon^\beta\bar\lambda_2\sum_{i=1}^n \xi_i^2 \ge \sum_{i,j=1}^n b_{ij}(x,t,\varepsilon)\xi_i\xi_j \ge \varepsilon^\beta\lambda_2\sum_{i=1}^n \xi_i^2, \]
\[ \bar\lambda_2\ge\lambda_2>0. \tag{11} \]

For problem (1), (3), (5), we assume, in addition, that in \(Q^+\)
\[ |\partial a/\partial x_k|\le K\varepsilon^\alpha,\quad |\partial b_{ij}/\partial x_k|\le K\varepsilon^\beta,\quad |\partial c_i/\partial x_k|\le K\varepsilon^\gamma, \]
\[ |\partial d/\partial x_k|\le K\varepsilon^\delta,\quad |\partial\varphi/\partial x_k|\le K\varepsilon^\eta. \tag{12} \]

For problem (1), (3), (6) the coefficients of equation (1) are extended to \(Q^+\) so that inequalities (9), (10), (11) are satisfied. In doing so we put
\[ d(x,t,\varepsilon)=-F^2(x,t)\varepsilon^{-\beta};\qquad b_{ij}(x,t,\varepsilon)=\varepsilon^\beta\quad \text{for } i=j; \]
\[ b_{ij}(x,t,\varepsilon)=0\quad \text{for } i\ne j. \tag{13} \]

By \(A^h, B^h_{ij}, C^h_i, D^h, \Phi^h\) we denote the averages of the thus obtained discontinuous functions \(a,b_{ij},c_i,d,\varphi\) with averaging radius \(h=\varepsilon^{|\delta|}\) for problem (1), (3), (4), \(h=\varepsilon^{|\beta|}\) for problem (1), (3), (5), and \(h=\varepsilon^\mu\), \(\mu>|\beta|\), for problem (1), (3), (6).

In the case of problem (1), (3), (5) the averaging is carried out in a special manner. We cover \(\overline Q^{-}\) by a finite number of domains \(R_i\) \((i=1,2,\ldots,N)\) such that in \(R_k\) containing points \(S=\{\Gamma\times[0T]\}\) it is possible to pass to coordinates \(t,y^k_1,\ldots,y^k_n\), \(y^k_i=\varphi_i(x_1,\ldots,x_n)\) \((i=1,2,\ldots,n)\), in which a piece of the boundary \(S\) is transformed into a piece of the plane \(y^k_n=0\), and in \(R_\ell\cap R_k\)
\[ y^k_n=y^\ell_n. \]
Denote
\[ R_{N+1}=\{E_n\times[0T]\}-\sum_{i=1}^N R_i. \]
Let \(\chi_k(x)\) be smooth functions, \(\chi_k(x)=0\) outside \(R_k\), \(0\le\chi_k\le1\),
\[ \sum_{k=1}^{N+1}\chi_k=1. \]
Put:
\[ A^h=\sum_{k=1}^{N+1}\chi_k(a)^h_k,\qquad B^h_{ij}=\sum_{k=1}^{N+1}\chi_k(b_{ij})^h_k,\qquad C^h_i=\sum_{k=1}^{N+1}\chi_k(c_i)^h_k, \]
\[ D^h=\sum_{k=1}^{N+1}\chi_k(d)^h_k,\qquad \Phi^h=\sum_{k=1}^{N+1}\chi_k(\varphi)^h_k, \]

where \((a)^h_k\), \((b_{ij})^h_k\), \((c_i)^h_k\), \((d)^h_k\), \((\varphi)^h_k\) denote the averages of the corresponding functions in the domain \(R_k\) with respect to the variables \(t, y^k_1, \ldots, y^k_n\).

Consider the Cauchy problem for the equation

\[ A^h(x,t,\varepsilon)\frac{\partial u}{\partial t} +\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i} \left(B^h_{ij}(x,t,\varepsilon)\frac{\partial u}{\partial x_j}\right) +\sum_{i=1}^{n} C_i^h(x,t,\varepsilon)\frac{\partial u}{\partial x_i} + \]

\[ + D^h(x,t,\varepsilon)u=\Phi^h(x,t,\varepsilon) \tag{14} \]

with the condition

\[ u(x,t)\big|_{t=0}=f^h(x), \tag{15} \]

where \(f^h(x)\in \overset{0}{C}{}^{(\infty)}(\Omega^-)\), \(f^h(x)=0\) outside \(\Omega^-\), and
\[ \|f^h(x)-f(x)\|_{\overset{0}{W}{}^1_2(\Omega^-)}\to 0 \]
as \(h\to 0\).

For the solution of problem (14), (15), estimates have been obtained, uniform in \(h\) and \(\varepsilon\), by means of which the following is proved.

Theorem 1. The solution \(u^h_\varepsilon\) of problem (14), (15) converges as \(\varepsilon\to 0\) in the mean in the domain \(Q^-_\sigma\), for any \(\sigma>0\), to the generalized solution \(u_0\) of problem (1), (3), (4), if \(\delta\leqslant \alpha\), \(\beta<-\delta\), \(\delta<0\); to the generalized solution of problem (1), (3), (5), if \(\delta\leqslant \alpha\), \(\beta>-\delta\), \(\beta>0\); to the generalized solution of problem (1), (3), (6), if \(-\beta<\alpha\), \(\beta=-\delta\), \(\beta>0\);

1. Suppose that in the cylinder \(Q^-\) there is given the hyperbolic equation

\[ a(x,t)\frac{\partial^2 u}{\partial t^2} +\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i} \left(b_{ij}(x,t)\frac{\partial u}{\partial x_j}\right) +\sum_{i=1}^{n} c_i(x,t)\frac{\partial u}{\partial x_i} + \]

\[ +g(x,t)\frac{\partial u}{\partial t}+d(x,t)u=\varphi(x,t). \tag{16} \]

The functions \(a, b_{ij}, c_i, g, d, \varphi, \partial a/\partial t, \partial b_{ij}/\partial t, \partial d/\partial t\) \((i,j=1,2,\ldots,n)\) are bounded and measurable; \(a(x,t)\) is a continuous function. The inequalities (2) are satisfied, \(S\subset A^2\). For equation (16) consider mixed problems with the initial conditions:

\[ u(x,t)\big|_{t=0}=f_0(x),\qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=f_1(x), \tag{17} \]

where \(f_0(x)\in \overset{0}{W}{}^1_2(\Omega^-)\), \(f_1(x)\in L_2(\Omega^-)\), and 1) the boundary condition of the first kind (4); 2) the boundary condition of the second kind (5); 3) the boundary condition of the third kind (6).

For problem (16), (17), (5), the functions \(\partial a/\partial x_k\), \(\partial b_{ij}/\partial x_k\), \(\partial c_i/\partial x_k\), \(\partial g/\partial x_k\), \(\partial d/\partial x_k\), \(\partial\varphi/\partial x_k\), \(\partial c_i/\partial t\), \(\partial g/\partial t\), \(\partial\varphi/\partial t\), \(\partial^2 b_{ij}/\partial t^2\) \((i,j,k=1,2,\ldots,n)\) are also bounded and measurable, \(S\subset A^3\), \(f_0(x)\in \overset{0}{W}{}^2_2(\Omega^-)\), \(f_1(x)\in \overset{0}{W}{}^1_2(\Omega^-)\), where \(\overset{0}{W}{}^2_2(\Omega^-)\) is the space obtained by completing the functions of the class \(C_0^{(\infty)}(\Omega^-)\) in the norm \(W^2_2\).

By a generalized solution of problem (16), (17), (4) we shall mean a function \(u(x,t)\in \overset{0}{W}{}^{1,1}_2(Q^-)\) such that \(u(x,0)=f_0(x)\) and, for any function \(\Phi(x,t)\in \overset{0}{W}{}^{1,1}_2(Q^-)\), \(\Phi(x,T)=0\), the integral identity holds:

\[ \widetilde{\Psi}(u,\Phi)\equiv \iint_{Q^-}\left[ -a\frac{\partial u}{\partial t}\frac{\partial\Phi}{\partial t} -\sum_{i,j=1}^{n} b_{ij}\frac{\partial u}{\partial x_j}\frac{\partial\Phi}{\partial x_i} +\sum_{i=1}^{n} c_i\frac{\partial u}{\partial x_i}\Phi -\frac{\partial a}{\partial t}\frac{\partial u}{\partial t}\Phi +\right. \]

\[ \left. +g\frac{\partial u}{\partial t}\Phi +du\Phi-\varphi\Phi \right]\,dx\,dt -\int_{\Omega^-}a(x,0)\Phi(x,0)f_1(x)\,dx=0. \tag{18} \]

By a generalized solution of problem (16), (17), (5) we shall mean a function \(u(x,t)\in W^{1,1}_2(Q^-)\) such that \(u(x,0)=f_0(x)\) and, for any function \(\Phi(x,t)\in W^{1,1}_2(Q^-)\), \(\Phi(x,T)=0\), the integral identity (18) is satisfied.

By a generalized solution of problem (16), (17), (6) we shall mean a function \(u(x,t)\in W^{1,1}_2(Q^-)\) such that \(u(x,0)=f_0(x)\) and, for any function \(\Phi(x,t)\in\)

\(\in W_2^{1,1}(Q^-)\), \(\Phi(x,T)=0\), the integral identity

\[ \widetilde{\Psi}(u,\Phi)-\int_S Fu\Phi\,ds=0. \tag{19} \]

We extend the functions \(a, b_{ij}, c_i, g, d, \varphi\) to \(Q^+\) for problem (16), (17), (4) so that inequalities (9), (11) and

\[ |g(x,t,\varepsilon)|\leqslant K\varepsilon^\omega, \tag{20} \]

are satisfied; for problem (16), (17), (6), conditions (9), (11), (13), (20); for problem (16), (17), (5), inequalities (9), (11), (12), (20) and

\[ |\partial^2 b_{ij}/\partial t^2|\leqslant K\varepsilon^\beta,\quad |\partial c_i/\partial t|\leqslant K\varepsilon^\gamma,\quad |\partial g/\partial t|\leqslant K\varepsilon^\omega, \]
\[ |\partial\varphi/\partial t|\leqslant K\varepsilon^\eta,\quad |\partial g/\partial x_i|\leqslant K\varepsilon^\omega \quad (i,j=1,2,\ldots,n). \]

\(A^h, B_{ij}^h, C_i^h, G^h, D^h, \Phi^h\) denote the averages of the corresponding discontinuous functions \(a, b_{ij}, c_i, g, d, \varphi\) with averaging radius: \(h=\varepsilon^{|\delta|}\) for problem (16), (17), (4); \(h=\varepsilon^{|\beta|}\) for problem (16), (17), (5); \(h=\varepsilon^\mu\), \(\mu>|\beta|\), for problem (16), (17), (6). In the case of problem (16), (17), (5), the averaging is carried out as for problem (1), (3), (5).

The exponents of powers of \(\varepsilon\) satisfy the conditions

\[ \gamma\geqslant(\beta+\alpha)/2,\quad \eta\geqslant\alpha/2,\quad \beta>\alpha,\quad \omega\geqslant\alpha. \]

Consider the Cauchy problem for the equation

\[ A^h(x,t,\varepsilon)\frac{\partial^2 u}{\partial t^2} +\sum_{i,j=1}^n \frac{\partial}{\partial x_i} \left(B_{ij}^h(x,t,\varepsilon)\frac{\partial u}{\partial x_j}\right) +\sum_{i=1}^n C_i^h(x,t,\varepsilon)\frac{\partial u}{\partial x_i} \]
\[ +G^h(x,t,\varepsilon)\frac{\partial u}{\partial t} +D^h(x,t,\varepsilon)u=\Phi^h(x,t,\varepsilon) \tag{21} \]

with initial conditions

\[ u(x,t)\big|_{t=0}=f_0^h(x),\qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=f_1^h(x), \tag{22} \]

where

\[ f_0^h(x),\quad f_1^h(x)\in \overset{0}{C}{}^{(\infty)}(\Omega^-),\qquad f_0^h(x)=f_1^h(x)=0\quad \text{outside }\Omega^-; \]
\[ \|f_0^h(x)-f_0(x)\|_{\overset{0}{W}{}_{2}^{1}(\Omega^-)} +\|f_1^h(x)-f_1(x)\|_{L_2(\Omega^-)}\to 0 \quad \text{as } h\to 0. \]

Theorem 2. The solution \(u_\varepsilon\) of problem (21), (22) converges as \(\varepsilon\to0\), in the mean in the domain \(Q_\sigma\) for \(\sigma>0\), to the generalized solution \(u_0\) of problem (16), (17), (4), if \(\alpha\leqslant\delta\), \(\beta<-\delta\), \(\delta<0\); to the generalized solution of problem (16), (17), (6), if \(\alpha>-\beta\), \(\beta=-\delta\), \(\beta>0\); to the generalized solution of problem (16), (17), (5), if \(\alpha\leqslant\delta\), \(\beta>-\alpha\), \(\beta>0\) and

\[ \|f_0^h(x)-f_0(x)\|_{\overset{0}{W}{}_{2}^{2}(\Omega^-)} +\|f_1^h(x)-f_1(x)\|_{\overset{0}{W}{}_{2}^{1}(\Omega^-)} \to 0 \quad \text{as } h\to 0. \]

If \(\beta>0\), \(\alpha>0\), \(\gamma>0\), \(\delta>0\), \(\eta>0\), \(\omega>0\), then for problem (16), (17), (5) Theorem 2 is true under the condition that \(f_0(x)\in \overset{0}{W}{}_{2}^{1}(\Omega^-)\), \(f_1(x)\in L_2(\Omega^-)\).

The author expresses deep gratitude to Prof. O. A. Oleinik for assistance in writing this work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
15 XI 1963

CITED LITERATURE

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