UDC 517.949.2

MATHEMATICS

M. L. RASULOV

APPLICATION OF THE CONTOUR INTEGRAL METHOD TO THE SOLUTION OF MULTIDIMENSIONAL MIXED PROBLEMS FOR PARABOLIC SYSTEMS OF SECOND ORDER

(Presented by Academician I. N. Vekua on 12 XII 1969)

In the present note the author gives an application of the contour integral method (according to the scheme of Chapters VIII—IX of the book (¹)) to the solution of mixed problems for parabolic systems of second order containing, in the boundary condition, the derivative with respect to time in a domain \(D\) of three dimensions with boundary \(\Gamma\).

Consider the system

\[ M\left(t,\frac{\partial}{\partial t}\right)v = A(x)\Delta v+ \sum_{i=1}^{3} A_i(x)\frac{\partial v}{\partial x_i} + A_0(x)v+f(x,t) \tag{1} \]

with the boundary condition

\[ \lim_{x\to z} \left\{ [\alpha_0(z)+\alpha_1(z)M(t,\partial/\partial t)]\,dv(x,t)/dn_z + [\beta_0(z)+\alpha_1(z)\beta_1(z)M(t,\partial/\partial t)]v(x,t) \right\} = \psi(z,t),\quad z\in\Gamma, \tag{2} \]

and the initial condition

\[ v(x,0)=\Phi(x), \tag{3} \]

where \(M(t,\partial/\partial t)=b_0(t)\partial/\partial t+b_1(t)\); \(b_0(t)\), \(b_1(t)\) are functions defined on the interval \([0,\infty)\); \(A(x)\), \(A_i(x)\) \((i=0,1,2,3)\) are square matrices of order \(m\), defined in the three-dimensional domain \(D+\Gamma\).

It is assumed that the following conditions are fulfilled:

\(1^\circ\). In the domain \(D+\Gamma\) the roots \(\nu_i(x)\) \((i,\ldots,m)\) of the characteristic equation

\[ \det(A(x)+\nu E)=0 \]

have constant multiplicity and strictly negative real parts. In addition, the functions \(b_0(t)\), \(b_1(t)\) are continuous on the interval \([0,\infty)\), and there exist positive numbers \(\varepsilon>0\), \(C>0\), \(\sigma\ge 0\), \(C_1>0\) such that the inequalities

\[ 0<\varepsilon\le b_0^{-1}(t)\le Ce^{\sigma t},\quad |b_1(t)|\le C_1. \]

hold.

\(2^\circ\). The matrices \(\alpha_k(z)\), \(\beta_k(z)\) \((k=0,1)\) are continuous on \(\Gamma\), and for sufficiently large values of the complex parameter \(\lambda\) the matrix

\[ [\lambda_0(z)+\lambda^2\alpha_1(z)]^{-1} [\beta_0(z)+\lambda^2\alpha_1(z)\beta_1(z)] \]

is bounded for \(z\in\Gamma\) by a number independent of \(\lambda\); \(\Gamma\) is a Lyapunov surface; for \(y\in\Gamma\) the vector-function \(\psi(y,t)\) is an original in the sense of note (²).

\(3^\circ\). The vector-functions \(\Phi(x)\), \(\partial^k f(x,t)/\partial t^k\) \((k=0,1,2)\) have continuous derivatives with respect to all \(x_i\) \((i=1,2,3)\) in the domain \(D+\Gamma\) for \(t\ge 0\) and vanish in some boundary strip of the domain \(D\).

In notes (³,⁴) the existence of a positive \(\delta\) was proved such that in the domain \(R_\delta\) of values of \(\lambda\) satisfying the inequalities

\[ |\lambda|\ge R,\quad |\arg\lambda|\le \pi/4+\delta, \tag{\(\mathit{R}_\delta\)} \]

there exists a solution, analytic in \(\lambda\), of the corresponding spectral problem

\[ A(x)\Delta u+\sum_{i=1}^{3} A_i(x)\frac{\partial u}{\partial x_i}+\bigl(A_0(x)-\lambda^2\bigr)u=\Phi(x), \tag{4} \]

\[ \lim_{x\to z} B\bigl(z,d/dn_z,\lambda^2\bigr)u(x,\lambda)=\widetilde{\psi}(z,\lambda), \tag{5} \]

where \(\widetilde{\psi}(z,\lambda)\) is the analytic continuation of the vector function

\[ \int_{0}^{\infty}\exp[-\lambda^2t_1(t)+t_2(t)]\,t_1'(t)\psi(z,t)\,dt \]

to the whole domain \(R_\delta\);

\[ t_1(t)=\int_{0}^{t} b_0^{-1}(\tau)\,d\tau,\qquad t_2(t)=\int_{0}^{t} b_0^{-1}(\tau)b_1(\tau)\,d\tau. \]

Let \(R\) be a sufficiently large positive number. Denote by \(S\) an infinite open contour lying in the domain \(R_\delta\), sufficiently remote parts of which (lying outside a circle with center at the origin and sufficiently large radius) coincide with the continuations of the rays \(\arg\lambda=\pm(\pi/4+\delta)\).

With the aid of the estimates obtained in Notes \((^3,^4)\), for the solution of the spectral problem, following the scheme of Chapter IX of the book \((^1)\), the following assertions are proved:

Theorem 1. Under conditions \(1^\circ\)—\(3^\circ\) there exists a solution \(v^{(1)}(x,t,\psi)\) of problem (1), (3), in the case \(f(x,t)\equiv0\), \(\Phi(x)\equiv0\), representable in the form of a contour integral

\[ v^{(1)}(x,t,\psi)=\frac{1}{\pi\sqrt{-1}}\int_S \exp[\lambda^2t_1(t)-t_2(t)]\lambda u^{(1)}(x,\lambda,\widetilde{\psi})\,d\lambda, \tag{6} \]

where \(u^{(1)}(x,\lambda,\widetilde{\psi})\) is the solution of the spectral problem (4), (5) for \(\Phi(x)\equiv0\) in \(D\).

Theorem 2. Under the conditions of Theorem 1, if the vector functions \(\Phi(k)\), \(\partial^k f(x,t)/\partial t^k\) \((k=0,1,2)\) for \(t\in[0,\infty)\) are continuously differentiable once in the domain \(D+\Gamma\) and vanish in some boundary strip of the domain \(D\), then problem (1)—(3) for \(\psi(z,t)\equiv0\) on \(\Gamma\) has a solution \(v^{(2)}(x,t,\Phi,f)\), represented by the formula

\[ v^{(2)}(x,t,\Phi,f)=\frac{1}{\pi\sqrt{-1}}\int_S \lambda\,d\lambda\int_D G(x,\xi,\lambda)z(\xi,t,\lambda)\,dD_\xi, \tag{7} \]

where \(G(x,\xi,\lambda)\) is the Green matrix of the spectral problem (4), (5); \(z(x,t,\lambda)\) is the solution of the Cauchy problem

\[ M(t,\partial/\partial t)z-\lambda^2z=f(x,t),\qquad z(0)=\Phi(x). \]

It is easy to verify that

\[ z(\xi,t,\lambda)=\Phi(\xi)\exp[\lambda^2t_1(t)-t_2(t)]+ \]

\[ +\int_{0}^{t} b_0^{-1}(\tau)f(\xi,\tau)\exp\{\lambda^2[t_1(t)-t_1(\tau)]-[t_2(t)-t_2(\tau)]\}\,d\tau. \]

The integrals (6) and (7) can be calculated according to the scheme of Note \((^5)\), and then for \(v^{(1)}\), \(v^{(2)}\) we obtain a representation in the form of a series whose terms are expressed through the data of the problem.

Azerbaijan State University
named after S. M. Kirov
Baku

Received
12 XII 1969

CITED LITERATURE

\(^1\) M. L. Rasulov, The Contour Integral Method, “Nauka,” 1964.
\(^2\) M. L. Rasulov, I. S. Zeinalov, DAN, 189, No. 5 (1969).
\(^3\) M. L. Rasulov, DAN, 192, No. 6 (1970).
\(^4\) M. L. Rasulov, DAN, 192, No. 5 (1970).
\(^5\) M. L. Rasulov, DAN, 128, No. 3 (1959).