UDC 517.946.9 : 536.2

THE STRUCTURE OF SOLUTIONS OF ELLIPTIC-PARABOLIC EQUATIONS WITH ONE SEPARABLE VARIABLE

V. B. RYVKIN

Problems concerning the joint action of convective and conductive transfer (“conjugate” problems) usually fall under the conditions of Fichera’s work [1], where questions of the formulation of well-posed problems for this type of equation are studied in detail. In the present work we investigate the structure of solutions of such problems in the case when separation of one of the variables is admissible under certain additional conditions. This yields a generalization of the corresponding results for elliptic and parabolic equations.

We shall carry out the consideration from the point of view of the theory of operators in a Hilbert space, sometimes leaving aside questions concerning the domains of definition of the operators; since the operators are considered up to closure, all assertions concern only weak solutions.

We consider equations of the form

\[ -\sum_{i,j}\frac{\partial}{\partial x_i} \left(\lambda_{ij}\frac{\partial u}{\partial x_j}\right) +\sum_i v_i\frac{\partial u}{\partial x_i} +cu=f, \tag{1} \]

where \(\lambda_{ij}\) is a symmetric nonnegatively definite matrix for all \(x\) under consideration, and it is assumed that the equation is defined in the cylinder \(\Omega \times l\), i.e. \((x_1,\ldots,x_{n-1})\in\Omega\), \(x_n\in(a,b)\). In order that separation of variables be possible, it is necessary to assume that \(\lambda_{ij}\), \(v_i\), and \(c\) are independent of \(x_n\). Furthermore, in order that separation of variables yield corresponding results for purely parabolic or degenerate elliptic equations (in the absence of convection), one must set \(\lambda_{in}\equiv0\), \(i\ne n\). We shall restrict ourselves to the case when \(v_i\equiv0\), \(i\ne n\). This restriction considerably narrows the class of problems under consideration, while at the same time making it possible to obtain information about a rather simple behavior of the solutions. Under the assumptions made, one may suppose that \(\lambda_{ij}\) \((ij=1,\ldots,n-1)\) do not simultaneously vanish in any subdomain \(\Omega_1\subset\Omega\), since otherwise it would be sufficient to consider the equation separately in \(\Omega_1\times l\), where it reduces to an ordinary differential equation in \(x_n\), while \(x_1,\ldots,x_{n-1}\) occur as parameters, and in the equation in \((\Omega\setminus\Omega_1)\times l\), where homogeneous conditions of the second kind must be imposed on the common part of the boundaries of \(\Omega\setminus\Omega_1\) and \(\Omega_1\).

We shall also impose the following restriction on \(\lambda_{ij}\) \((ij=1,\ldots,n-1)\) and boundary conditions on the boundary \(\Gamma\) of the domain \(\Omega\): the operator

\[ Lu=-\sum_{i,j=1}^{n-1}\frac{\partial}{\partial x_i}\times \]

\[ \times\left(\lambda_{ij}\frac{\partial u}{\partial x_j}\right)+cu \]

is self-adjoint under these boundary conditions.

by a strictly positive definite operator in the Hilbert space \(L_2(\Omega)\) (here a corresponding choice of the domain of definition is also meant).

Starting from this, equation (1) can be written in operator form

\[ Lu - B\,\frac{\partial^2 u}{\partial x_n^2} + C\,\frac{\partial u}{\partial x_n}=f. \tag{2} \]

Here \(u\) has the meaning of a vector-valued function on the interval \(l\) with values in \(L_2(\Omega)\), \(B\) is the operator of multiplication by \(\lambda_{nn}\), and \(C\) is the operator of multiplication by \(v_n\).

In accordance with the results of [1], for equation (2) at \(x_n=a\) and \(x_n=b\) (\(x_n=x\)) one may impose the following boundary conditions (we consider only the case of conditions of the first kind): at \(x=a\) one should prescribe the projection of \(u(a)\) onto \(L_2(\Omega_B\cup\Omega_{C+})\), and at \(x=b\) the projection of \(u(b)\) onto \(L_2(\Omega_B\cup\Omega_{C-})\), where \(\Omega_B\) is the subdomain of \(\Omega\) where \(\lambda_{nn}>0\), while \(\Omega_{C+}\) and \(\Omega_{C-}\) are respectively the subdomains of \(\Omega\) where \(v_n>0\) and \(v_n<0\). This assertion is obviously generalized to the case of permutable \(B\) and \(C\); then (2) should be understood as an equation for a vector-valued function on \((a,b)\) with values in some Hilbert space \(H\), \(L\) a strictly positive definite self-adjoint operator, \(B\) a nonnegative self-adjoint operator, and \(C\) a self-adjoint operator. At \(x=a\) one should prescribe the projection of \(u(a)\) onto the sum of the subspaces \([E-E(+0,B)]H\) and \([E-E(+0,C)]H\); at \(x=b\), respectively, \([E-E(+0,B)]H\) and \(E(-0,C)H\), where \(E(\lambda,A)\) is the spectral function of the operator \(A\).

In the case of nonpermutable \(B\) and \(C\), an analogous circumstance holds with the following change: in place of the operator \(C\) in the formulation given above one should take \(E(+0,B)CE(+0,B)\). In doing so, one must assume that the indicated product gives a self-adjoint operator upon closure.

We shall show that for homogeneous boundary conditions at \(x=a,b\) in the last case, uniqueness of solutions is ensured, and their existence for arbitrary \(f\).

To prove uniqueness of the solution, i.e., that when \(f\equiv0\) one also has \(u\equiv0\), multiply (2) scalarly by \(u\) and integrate from \(a\) to \(b\) with respect to \(x=x_n\):

\[ \int_a^b (Lu,u)\,dx-\int_a^b (Bu_{xx},u)\,dx+\int_a^b (Cu_x,u)\,dx \]

\[ =\int_a^b (Lu,u)\,dx+\int_a^b (Bu_x,u_x)\,dx- \]

\[ -\left.(Bu_x,u)\right|_a^b+\left.\frac{1}{2}(Cu,u)\right|_a^b=0. \tag{3} \]

Here integration by parts is performed on the expression \((Bu_{xx},u)\) and the expression \((Cu_x,u)\) is integrated; treating \(B\) and \(C\) as scalars is possible in view of their commutativity with the operation of differentiation with respect to \(x\), since \(B\) and \(C\) do not depend on \(x\).

From (4) it is seen that the nonintegral terms are nonnegative, and in view of the nonnegativity of \(B\) and the strict positivity of \(L\), the whole expression can vanish only if \(u=0\) for almost all \(x\). Thus uniqueness is proved. To prove existence of a solution, suppose that

that \(f\) is measurable and square-summable on the interval \((a,b)\). We shall show that the operator
\[ \Lambda=L-B\frac{\partial^2}{\partial x^2}+C\frac{\partial}{\partial x}, \]
defined on vector-functions twice continuously differentiable with respect to \(x\) on \((a,b)\) with values in \(D_L\cap D_B\cap D_C\) and satisfying homogeneous boundary conditions, admits a closure, i.e., there does not exist a sequence such that \(\Lambda u_n\to u_0\ne 0,\ u_n\to 0\).

Consider the operator \(\Lambda^*\), formally adjoint to \(\Lambda\) in the Hilbert space \(H(a,b)\) of measurable square-summable vector-functions with values in \(H\), i.e., in fact in \(\int_a^b \oplus H\,dx\) (see [2]). It differs from \(\Lambda\) only by the change of sign before \(C\), with the corresponding change in the specification of the boundary conditions, where \(C\) must likewise be replaced by \(-C\). It is, obviously, defined at least on twice differentiable vector-functions with values in \(D_L\cap D_B\cap D_C\) and with the corresponding homogeneous boundary conditions at \(x=a\) and \(x=b\). Since \((\Lambda u,v)=(u,\Lambda^*v)\), \(u\in D_\Lambda,\ v\in D_{\Lambda^*}\), it follows that if \(D_{\Lambda^*}\) is dense in \(H(a,b)\), then we can approximate \(u_0\) by some \(v_0(\varepsilon)\in D_{\Lambda^*}\) so that \(\|u_0-v_0(\varepsilon)\|<\varepsilon\). Then we have
\[ (\Lambda u_n,v_0)=(u_n,\Lambda^*v_0). \]
Passing to the limit, we have \((u_0,v_0(\varepsilon))=0\). Together with \(\|u_0-v_0(\varepsilon)\|<\varepsilon\), this gives \(u_0=0\). Thus the operator \(\Lambda\) admits a closure if \(D_L\cap D_B\cap D_C\) is dense in \(H\). In order that the adjoint operator \(\Lambda^*\) have an analogous structure with \(C\) replaced by \(-C\), it is necessary to require that the restrictions of the operators \(L,B,C\) to \(D_L\cap D_B\cap D_C\), upon closure, give the original operators. This, in particular, is fulfilled if \(B\) and \(C\) are bounded operators. With an analogous structure of \(\Lambda\) and \(\Lambda^*\) we shall also have an existence theorem. We note that if the restriction of the operators \(L,B\), and \(C\) to the above-indicated domain is not restored to the original operators by closure, then there is a solution for some closed subspace \(H'(a,b)\subseteq H(a,b)\) for \(f\).

Let us summarize the result obtained.

Theorem 1. If the conditions are fulfilled: 1) \(L>c>0\); 2) \(B\geqslant 0,\ C=C^*\); 3) \(L,B\), and \(C\) are restored from their values on \(D_L\cap D_B\cap D_C\), then problem (2) with boundary conditions
\[ P_a u(a)=\{E-E(+0,B)+E-- \]
\[ -E[+0,E(+0,B)CE(+0,B)]\}u(a)=0, \]
\[ P_b u(b)=\{E-E(+0,B)+E[-0,E(+0,B)CE(+0,B)]\}u(b)=0 \tag{4} \]
(which are assumed to be well-defined, i.e., the spectral function is taken from a self-adjoint operator, and the domains of definition of \(\Lambda\) and \(\Lambda^*\) have, in the sections \(x=a\) and \(x=b\), sets dense in the subspaces subordinate to the corresponding conditions of type (4)) has, for arbitrary \(f\in H(a,b)\), a unique solution; moreover, the adjoint problem differs from the original one by the change of sign of \(C\) in (2) and (4); for it the assertion on uniqueness and existence of the solution is also valid.

A generalization of this theorem to the case where \(L,B\), and \(C\) depend on \(x\) can be outlined as follows: replace equation (2) by the equation
\[ \left(L(x)-\frac{d}{dx}B(x)\frac{d}{dx}+C\frac{d}{dx}\right)u=f. \]

In conditions (3), the operators should be taken respectively at \(x=a\) and \(x=b\), and for the theorem to be valid one must require

\[ L \pm \frac{1}{2}\frac{dC}{dx} \gg 0. \]

Another generalization of Theorem 1 concerns the non-self-adjointness of \(L\) under the condition that \(L+L^*\) is a strictly positive self-adjoint operator, which covers the case \(v_i\ne 0,\ i\ne n\).

Let us also point out one essential circumstance, restricting ourselves to the case in which the operators are independent of \(x\). The homogeneity of the equation with respect to \(x\) makes it possible to introduce spaces of basic and generalized vector-functions of \(x\) with values in \(H\). Generalized functions of finite order can be introduced by differentiating with respect to \(x\) the initial functions \(u(x)\in H(a,b)\). In view of the commutativity of the operator of differentiation with respect to \(x\) with the operator \(\Lambda\) without boundary conditions, in particular, for functions equal to zero in some neighborhood of the points \(a\) and \(b\), one can define a solution for the case where \(f\) is a generalized function. Then, in particular, one can restrict oneself only to homogeneous conditions at \(x=a,\ x=b\).

We now pass to separation of variables. Formal separation of the variable \(x\) in the form \(u_\lambda(x)=e^{\lambda x}u_\lambda\) gives for \(\lambda\) the quadratic equation

\[ (Lu_\lambda,u_\lambda)+\lambda(Cu_\lambda,u_\lambda)-\lambda^2(Bu_\lambda,u_\lambda)=0, \tag{5} \]

which has real roots of different signs, and one of them, or both, may tend to \(\infty\).

The problem of determining \(u_\lambda\):
\(Lu_\lambda+\lambda Cu_\lambda-\lambda^2Bu_\lambda=0\)
is easily reduced to the form considered in [3, 4]. We shall dwell on the method used in [4]. Introduce a new quantity
\(v=B^{1/2}\dfrac{du}{dx}\), where \(B^{1/2}\) is the positive square root of the operator \(B\); then (2) is written

\[ \begin{pmatrix} L&0\\ 0&E \end{pmatrix} \binom{u}{v} = \frac{d}{dx} \begin{pmatrix} -C&B^{1/2}\\ B^{1/2}&0 \end{pmatrix}. \tag{6} \]

If \(B\) is nondegenerate, then we have only a doubling of the dimension of the space \(H\); for (6) the eigenvalue problem

\[ \begin{pmatrix} L&0\\ 0&E \end{pmatrix} \binom{u_\lambda}{v_\lambda} = \lambda \begin{pmatrix} -C&B^{1/2}\\ B^{1/2}&0 \end{pmatrix} \binom{u_\lambda}{v_\lambda} \tag{7} \]

is an ordinary symmetrizable problem of the type \(Au=\lambda A_1u\) with positive definite \(A\) and self-adjoint \(A_1\); for such a problem the existence of a spectral function is known, and in (7), for completeness of the system of eigenvectors, one should also take eigenvectors corresponding to \(\lambda=\infty\), if \(A_1\) is degenerate; the eigenvectors are orthogonal with weight \(A\), with the corresponding generalization to the case of a spectrum that is not purely discrete.

Consider problem (2) for \(a=-\infty,\ b=+\infty\) (the boundary conditions then are simply decay of the solution at \(\infty\)) with \(f=\delta(x)u\); the case of arbitrary \(f\) is then obtained by integration.

As for problem (6), for \(f=\delta(x)u\), we shall seek its solution in the form

\[ \int_{\lambda<0} u_\lambda e^{\lambda x}\,d\sigma(\lambda),\quad x>0,\qquad \int_{\lambda>0} u_\lambda e^{\lambda x}\,d\sigma(\lambda),\quad x<0. \]

Substitution into (6) leads to the problem of representing the vector \(\binom{f}{0}\) in the form

\[ -\int \frac{1}{\lambda}\binom{Lu_\lambda}{v_\lambda}\operatorname{sgn}\lambda\,d\sigma(\lambda); \]

the latter problem is solvable by using the completeness and orthogonality with weight

\[ A=\begin{pmatrix}L&0\\0&E\end{pmatrix} \]

of the system \(\left\{\binom{u_\lambda}{v_\lambda}\right\}\).

Thus we can find the fundamental solution of problem (2)—(4) for \(a=-\infty,\ b=+\infty\). Having such a potential, we can construct the solution of problem (2) without imposing the conditions (4). To find the solution of problem (2)—(4), we must prescribe sources of the form

\[ \binom{\delta(x-a-0)f_{1a}}{\delta(x-a-0)f_{2a}} \]

and

\[ \binom{\delta(x-b+0)f_{1b}}{\delta(x-b+0)f_{2b}} \]

such that the conditions (4) are satisfied. It is verified directly that for the representation of \(f_a\) and \(f_b\) in terms of eigenfunctions we obtain the equations

\[ \binom{f_{1a}}{f_{2a}}=\int \binom{u_\lambda}{v_\lambda}\,d\sigma_a(\lambda),\qquad \binom{f_{1b}}{f_{2b}}=\int \binom{u_\lambda}{v_\lambda}\,d\sigma_b(\lambda); \]

\[ P_a\left[ \int_{\lambda<0} u_\lambda\,d\sigma_a(\lambda) -\int_{\lambda>0} u_\lambda e^{-\lambda(b-a)}\,d\sigma_b(\lambda) -u_0(a) \right]=0, \]

\[ P_b\left[ -\int_{\lambda>0} u_\lambda\,d\sigma_b(\lambda) +\int_{\lambda<0} u_\lambda e^{\lambda(b-a)}\,d\sigma_a(\lambda) +u_0(b) \right]=0. \tag{8} \]

Here \(u_0(x)\) is the solution obtained without using boundary conditions, if (2) and (6) are regarded as equations for \(-\infty<x<+\infty\) and \(f\) is continued in some way, for example by zero, outside the interval \((a,b)\). From (8) it is clear that \(\sigma_a(\lambda)\) is essential only for \(\lambda<0\), and \(\sigma_b(\lambda)\) only for \(\lambda>0\).

Thus the solution of the original problem (2)—(4) has been reduced to solving the system (8) with respect to \(\sigma_a(\lambda)\), \(\lambda<0\), and \(\sigma_b(\lambda)\), \(\lambda>0\). The uniqueness of the solution of system (8) follows from the uniqueness of the solution of problem (2)—(4). It should be noted that only \(\sigma_a(\lambda)\) for \(\lambda<0\) and \(\sigma_b(\lambda)\) for \(\lambda>0\) are determined uniquely; the other components of the end sources do not affect the solution. To prove the existence of a solution of (8), one may use the fact that a solution of this problem exists. Indeed, let us continue the solution of problem (2)—(4) by zero outside the interval \((a,b)\) and compute what addition to \(f\), in the form of generalized functions, ensures such a solution for \(-\infty<x<+\infty\).

By direct substitution into (2), using (4), we see that this addition is equal to

\[ \left(Cu(a+0)-B\frac{\partial u}{\partial x}\right)\delta(x-a-0) \]

\[ -Bu(a+0)\delta'(x-a-0) -\left(Cu(b-0)-B\frac{\partial u}{\partial x}\right)\delta(x-b+0)+ \]

\[ +Bu(b-0)\delta'(x-b+0). \]

For (6) this means that one may take

\[ \begin{pmatrix} f_{1a}\\ f_{2a} \end{pmatrix} = - \left( \begin{array}{cc} -C & B^{1/2}\\ B^{1/2} & 0 \end{array} \right) \begin{pmatrix} u\\ v \end{pmatrix} \bigg|_{x=a+0}, \]

\[ \begin{pmatrix} f_{1b}\\ f_{2b} \end{pmatrix} = + \left( \begin{array}{cc} -C & B^{1/2}\\ B^{1/2} & 0 \end{array} \right) \begin{pmatrix} u\\ v \end{pmatrix} \bigg|_{x=b-0}. \]

We shall not dwell here on questions of the interpretation and justification of these equalities and of the actual solution of system (8). Let us note that, when finite-dimensional approximations of the operators \(L\), \(B\), and \(C\) are used, no complications arise in solving systems of type (8), since they are finite. In some cases it is possible to investigate the infinite system of type (8); for example, the method considered in [5] carries over to the case of one-dimensional \(\Omega\) and finite-dimensional \(C\). In all these constructions the fundamental fact is the completeness of the system of projections of the eigenfunctions \(P_b u_\lambda\) with positive \(\lambda\) onto the subspace—the range \(P_b H\); and, for negative \(\lambda\), respectively, \(P_a u_\lambda\), the projections of \(u_\lambda\) onto the subspace of values \(P_a H\). The latter follows from the solvability of problem (2)—(4), respectively, for \(a=-\infty\) and \(b=+\infty\).

References

1. Fischer G. Mathematics (collection of translations), 7 : 6, 1963, pp. 99—121.
2. Naimark M. A., Fomin S. V. Uspekhi Mat. Nauk, vol. X, 2 (64), 111—142, 1955.
3. Kharazov D. F. Mat. sb., 42 (84), 2, 1957, pp. 129—178.
4. Müller P. H. Archiv der Mathematik, V. XII, F. 4, 1961, 307—310.
5. Rybkin V. B., Kondrashov N. G. IFZh, 6, 5, 92—98, 1963.

Received by the editors
March 14, 1965

Institute of Heat and Mass Transfer
Academy of Sciences of the BSSR