UDC 517.947.42

ON THE SMOOTHNESS OF THERMAL POTENTIALS. IV

APPLICATIONS OF THE THEORY OF THERMAL POTENTIALS TO THE SOLUTION OF A BIOPHYSICAL PROBLEM ON THE DISTRIBUTION OF CONCENTRATIONS IN A LIVING CELL

L. I. Kamynin

The paper considers applications of the theory of the special Panny thermal potential, constructed in article [4] of the author’s series of papers [1]—[4], to a boundary-value problem for a system of parabolic equations with discontinuous coefficients, arising in biophysics (see [5]) in the study of the question of the distribution of concentrations of substances participating in the vital processes of a living cell. The boundary conditions and the conjugation conditions on the surface of discontinuity of this problem include derivatives in oblique directions. The study of the smoothness of the special thermal potential of a Panny simple layer [6], depending on the smoothness of its density, defined on noncylindrical surfaces of type \(J_{1}^{0,1,(1+\alpha)/2}\) and \(J_{1}^{1,\alpha,\alpha/2}\), carried out in [4], makes it possible to prove the existence of a solution from the class \(H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\) (in the regions of continuity of the solution) for our problem under the minimally admissible smoothness requirements on the data of the problem. (These requirements do not differ from the smoothness requirements under which the existence of a solution from the class \(H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\) is ensured for the boundary-value problem with an oblique derivative for a second-order parabolic equation with continuous coefficients (see [4])). The proof of the existence theorem is carried out by the classical (see [7]) method of continuation with respect to a parameter, using a \((2+\alpha)\) a priori estimate of Schauder type, established for the solution of the boundary-value problem under consideration. (The derivation of this estimate is carried out by the method of papers of the author and V. N. Maslennikova [8]—[10], who obtained a \((2+\alpha)\) a priori estimate for the solution of the second and third boundary-value problems with oblique derivative in noncylindrical domains for a second-order parabolic equation.) Vyborny’s theorem ([11], [12]) on the sign of the oblique derivative of a solution of a parabolic equation at a boundary point of extremum, as well as studies by the author and V. N. Maslennikova ([13], [14]) on applications of the maximum principle to parabolic equations with discontinuous coefficients, make it possible to indicate conditions under which the solution of our problem is unique, and also admits an a priori estimate of the modulus.

The article consists of 6 sections. It uses the notation and definitions of papers [2]—[4], while preserving a single numbering of sections for the entire series. In § 17 the model problem (cf. [5]) on the distribution of concentrations of substances in a living cell is formulated, and a theorem on the existence and uniqueness of this problem is given. In § 18 the formulation is given of the general boundary-value problem, in various modifications, for a parabolic system preserving the specific properties of the model system from § 17. In § 19 the uniqueness of the solution of the boundary-value problem for a parabolic weakly coupled system is proved. In § 20, for a somewhat narrower class of parabolic weakly coupled systems, a theorem on an a priori ...

estimate of the modulus of the solution. In § 21 a theorem is proved on the existence, in the class \(H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\), of a solution of the auxiliary problem for a system of parabolic equations with piecewise constant coefficients. In § 22 a \((2+\alpha)\) a priori estimate of Schauder type is derived for the solution of the general boundary-value problem of § 18, and with its aid a theorem is proved on the existence of a solution of this problem in the class \(H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\).

The formulation of the results of the present work is contained in the author’s note [15].

§ 17. ON THE PROBLEM OF THE DISTRIBUTION OF CONCENTRATIONS IN A LIVING CELL

The study of a living cell leads to the necessity of studying the distribution of the substrate entering the cell and of the excreted compounds inside and outside it. Following [5], we shall briefly outline the biophysical picture of this process.

The substrate \(S_1\) entering the cell supplies the organism with energy and, in subsequent enzymatic transformations, forms low-molecular-weight substances \(S_2\) excreted by the cell; moreover, from this same substrate \(S_1\) the cell synthesizes compounds specific to it. Let \(u_{11}\) and \(u_{12}\) be the concentrations of the substrate \(S_1\) inside the cell and in the external medium, respectively. Similarly, let \(u_{21}\) and \(u_{22}\) be the corresponding concentrations of the excreted substances \(S_2\). If \(u_3\) is the concentration of enzymes, averaged over the period of division and the volume of the cell, then we introduce constants \(A>0\) and \(B>0\), where \(A=A_{12}u_3\) is the energy function of the cell, characterizing the ability of the biomass enzymes to transform the substrate \(S_1\) into the products \(S_2\), and \(B=B_{13}u_3\) is the plastic function of the cell, characterizing the ability of the biomass enzymes to transform the substrate \(S_1\) into substances specific to the cell. Further, let \(d_{i1}>0\) and \(d_{i2}>0\) \((i=1,2)\) be the diffusion coefficients of the \(i\)-th substance inside and outside the cell, respectively. Finally, by \(h_i>0\) \((i=1,2)\) we denote the permeability of the cell membrane \(\Gamma^{(1)}\) for the substance \(S_i\). The total reaction of the substrate \(S_1\) with the enzymes, taking diffusion into account, leads to the following problem for a system of parabolic equations:

\[ \left\{ \begin{aligned} \frac{\partial u_{11}}{\partial t} &= d_{11}\Delta u_{11}-(A+B)u_{11},\\ \frac{\partial u_{21}}{\partial t} &= d_{21}\Delta u_{21}+Au_{11}, \end{aligned} \right. \qquad (x,t)\in D_T^{(1)}, \]

\[ \frac{\partial u_{k2}}{\partial t}=d_{k2}\Delta u_{k2}, \qquad (x,t)\in D_T^{(2)} \qquad (k=1,2) \tag{17.1} \]

with initial conditions

\[ u_{kl}(x,0)=f_{kl}^{(1)}(x), \qquad x\in \overline{\Omega}^{(l)}=\overline{D_T^{(l)}\cap\{t=0\}} \qquad (k,l=1,2), \tag{17.2} \]

with boundary conditions on \(\Gamma^{(2)}\), the boundary of the external medium \(D_T^{(2)}\),

\[ \frac{\partial u_{k2}(x,t)}{\partial N^{(22)}(x,t)} -b_k(x,t)u_{k2}(x,t) =f_k^{(2)}(x,t), \qquad (x,t)\in \Gamma^{(2)} \quad (k=1,2) \tag{17.3} \]

under the conjugation conditions on \(\Gamma^{(1)}\), the surface of the cell membrane,

\[ -d_{k1}\frac{\partial u_{k1}(x,t)}{\partial N^{(11)}(x,t)} =h_k\bigl(u_{k2}(x,t)-u_{k1}(x,t)\bigr)= \]

\[ = d_{k2}\frac{\partial u_{k2}(x,t)}{\partial N^{(12)}(x,t)},\qquad (x,t)\in \Gamma^{(1)}\quad (k=1,2). \tag{17.4} \]

Here \(D_T=D_T^{(1)}\cup \Gamma^{(1)}\cup D_T^{(2)}\) is a bounded domain of the space \((x,t)\), enclosed between the two planes \(t=0\) and \(t=T>0\), having as its lower base the domain \(\Omega=\Omega^{(1)}\cup \Gamma_0^{(1)}\cup \Omega^{(2)}\) and as its lateral boundary the closed surface \(\Gamma^{(2)}\); here \(D_T^{(1)}\) is the internal subdomain of the domain \(D_T\) with lateral boundary surface \(\Gamma^{(1)}\), separating \(D_T^{(1)}\) and \(D_T^{(2)}\). The closed surfaces \(\Gamma^{(l)}\) \((l=1,2)\) are situated between the hyperplanes \(t=0\) and \(t=T>0\), admit representations of the form (1.5) (with \(\psi_l\)) from § 1 [2], and do not intersect one another. \(N_{(x,t)}^{(sl)}\) \((l=1,2\) for \(s=1\) and \(l=2\) for \(s=2)\) is the direction of the interior (with respect to \(\Omega_t^{(l)}=D_T^{(l)}\cap\{\tau=t\}\)) normal to the section \(\Gamma_t^{(s)}\) at the point \((x,t)\in \Gamma_t^{(s)}\).

In §§ 19–22 we shall prove the following theorem on the existence and uniqueness of the solution of problem (17.1)—(17.4).

Theorem 22. Suppose that the following conditions are satisfied:

1) The surfaces \(\Gamma^{(l)}\) are of type \(\Pi_{1}^{1,\beta,\beta/2}\), \((0<\alpha<\beta\leqslant1)\), and \(\Gamma^{(1)}\) and \(\Gamma^{(2)}\) do not intersect one another.

2)
\[ f_{kl}^{(1)}(x)\in H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\bigl(\overline{\Omega}^{(l)}\bigr),\qquad b_k(x,t),\ f_k^{(2)}(x,t)\in H_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}\bigl(\Gamma^{(2)}\bigr), \]
and \(f_{kl}^{(1)}(x)\), \(b_k(x,t)\), \(f_k^{(2)}(x,t)\), \(d_{kl}\), and \(h_k\) are compatible by virtue of (17.1)—(17.4) on the edges \(\overline{\Omega}^{(l)}\cap\Gamma^{(s)}\) \((l=1,2\) for \(s=1\) and \(l=2\) for \(s=2)\). Then there exists a unique solution \(u_{kl}(x,t)\) \((k,l=1,2)\) of problem (17.1)—(17.4) in the class
\[ H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\bigl(\overline{D}_T^{(l)}\bigr). \]

Remark. If \(f_{kl}^{(1)}(x)\equiv0\), \(\Omega^{(1)}\) is a sphere, and \(\Omega\) is the whole space \((x)\), then the existence of a solution of problem (17.1), (17.2), (17.4), without investigating its smoothness, was proved in [5].

§ 18. FORMULATION OF A GENERAL BOUNDARY-VALUE PROBLEM FOR A PARABOLIC SYSTEM OF EQUATIONS WITH DISCONTINUOUS COEFFICIENTS, PRESERVING THE SPECIFIC FEATURES OF THE MODEL SYSTEM (17.1)—(17.4)

Instead of problem (17.1)—(17.4) we shall consider more general boundary-value problems for systems of parabolic equations with piecewise continuous coefficients that have discontinuities of the first kind when crossing the surface \(\Gamma^{(1)}\). Let \(D_T^{(l)}\) \((l=1,2)\) from § 17 be bounded domains of the Euclidean \((n+1)\)-dimensional space \((x,t)=(x_1,\ldots,x_n;t)\). Let the surfaces \(\Gamma^{(l)}\) not intersect one another, and suppose that for any points \((x,t)\in\Gamma^{(1)}\) and \((y,\tau)\in\Gamma^{(2)}\) the inequality
\[ r(x,y)+|t-\tau|\geqslant 2d>0,\qquad d\ \text{is a constant}. \tag{18.1} \]
holds.

Let fields of directions with direction cosines \(\nu_k^{(sl)}(x,t)\) be given on the surface \(\Gamma^{(s)}\) \((k=1,2,\ldots,m;\ l=1,2\ \text{for}\ s=1\ \text{and}\ l=2\ \text{for}\ s=2)\), lying in the section \(\Omega_t^{(l)}\) and forming acute angles with the normals \(N_{(x,t)}^{(sl)}\) interior with respect to \(\Omega_t^{(l)}\) at the point \((x,t)\in\Gamma_t^{(s)}\). The conditions
\[ \bigl(\nu_k^{(sl)}(x,t),\,N_{(x,t)}^{(sl)}\bigr)\geqslant 2d_0>0, \tag{18.2} \]
will always be assumed to hold, where
\[ k=1,2,\ldots,m,\quad (x,t)\in\Gamma_t^{(s)},\quad l=1,2\ \text{for}\ s=1\ \text{and}\ l=2\ \text{for}\ s=2, \]
and \(d_0\) is a constant.

We shall study the solutions \(u_{kl}(x,t)\) \((k=1,2,\ldots,m;\ l=1,2)\) of the following systems of parabolic equations \((r=1,2,3)\):

\[ \begin{aligned} L_{kl}^{(r)}(u_{kl}) \equiv \sum_{i,j=1}^{n} a_{ij}^{(kl)}(x,t)\frac{\partial^{2}u_{kl}}{\partial x_i\partial x_j} &+\sum_{i=1}^{m}\sum_{j=1}^{n} b_{ij}^{(kl)}(x,t)\frac{\partial u_{il}}{\partial x_j} \\ &+\sum_{i=1}^{m} c_i^{(kl)}(x,t)u_{il} -\frac{\partial u_{kl}}{\partial t} =f_{kl}(x,t);\quad (x,t)\in D_T^{(l)},\\ &\hspace{4.5em} k=1,2,\ldots,m;\quad l=1,2 . \end{aligned} \tag{18.3\(_r\)} \]

with initial conditions

\[ u_{kl}(x,0)=f_{kl}^{(1)}(x),\quad x\in \overline{\Omega}^{(l)},\quad k=1,2,\ldots,m;\quad l=1,2 \tag{18.4} \]

with boundary conditions on \(\Gamma^{(2)}\)

\[ \begin{aligned} l_k(u_{k2}) \equiv a_k^{(k)}(x,t)\frac{\partial u_{k2}(x,t)}{\partial \nu_k^{(22)}(x,t)} -b_k^{(k)}(x,t)u_{k2}(x,t) &= \\ =\sum_{\substack{i\ne k,\ i=1}}^{m}\left( -a_i^{(k)}(x,t)\frac{\partial u_{i2}(x,t)}{\partial \nu_i^{(22)}(x,t)} +b_i^{(k)}(x,t)u_{i2}(x,t)\right) &+ f_k^{(2)}(x,t),\\ &\hspace{-4em} k=1,2,\ldots,m;\quad (x,t)\in \Gamma^{(2)} \end{aligned} \tag{18.5\(_r\)} \]

provided that the conjugation conditions on \(\Gamma^{(1)}\) are fulfilled

\[ \begin{aligned} l_{kl}(u_{k1},u_{k2}) \equiv (-1)^l d_{kl}^{(k)}(x,t) \frac{\partial u_{kl}(x,t)}{\partial \nu_k^{(1l)}(x,t)} - h_{kl}^{(kl)}(x,t)\bigl(u_{k2}(x,t) &- \\ -u_{k1}(x,t)\bigr) = \sum_{\substack{i\ne k,\ i=1}}^{m} \Biggl[ (-1)^{l+1} d_{il}^{(k)}(x,t) \frac{\partial u_{il}(x,t)}{\partial \nu_i^{(1l)}(x,t)} +& \tag{18.6\(_r\)} \end{aligned} \]

\[ \begin{aligned} &\quad +\sum_{j=1}^{2} h_{ij}^{(kl)}(x,t)u_{ij}(x,t)\Biggr] + f_{kl}^{(3)}(x,t),\\ &\hspace{4em} k=1,2,\ldots,m;\quad l=1,2;\quad (x,t)\in \Gamma^{(1)} . \end{aligned} \]

For \(r=2,3\), in (18.3)—(18.6) the following conditions are assumed to be satisfied:

\[ \left\{ \begin{aligned} & b_{ij}^{(kl)}(x,t)\equiv c_i^{(kl)}(x,t)\equiv 0 \quad \text{in } (18.3_2),\\ & a_i^{(k)}(x,t)\equiv b_i^{(k)}(x,t)\equiv 0 \quad \text{in } (18.5_2),\\ & d_{il}^{(k)}(x,t)\equiv h_{ij}^{(kl)}(x,t)\equiv 0 \quad \text{in } (18.6_2)\ \text{for } i\ge k+1, \end{aligned} \right. \tag{18.7\(_2\)} \]

\[ \left\{ \begin{aligned} & b_{ij}^{(kl)}(x,t)\equiv 0\ \text{for } i\ne k,\quad c_i^{(kl)}(x,t)\equiv 0\ \text{for } i\ge k+1 \quad \text{in } (18.3_3),\\ & a_i^{(k)}(x,t)\equiv 0\ \text{for } i\ne k,\quad b_i^{(k)}(x,t)\equiv 0\ \text{for } i\ge k+1 \quad \text{in } (18.5_3),\\ & d_{il}^{(k)}(x,t)\equiv 0\ \text{for } i\ne k,\quad h_{ij}^{(kl)}(x,t)\equiv 0\ \text{for } i\ge k+1 \quad \text{in } (18.6_3). \end{aligned} \right. \tag{18.7\(_3\)} \]

In the systems (18.3\(_r\))—(18.6\(_r\)) we shall always assume the following conditions to be satisfied:

I. The equations are uniformly parabolic in \(\overline{D}_T^{(l)}\), i.e., for any real \(\lambda_1,\lambda_2,\ldots,\lambda_n\) and all \((x,t)\in \overline{D}_T^{(l)}\) there exist constants \(M_i>0\) \((i=0,1)\) such that

\[ M_1 \sum_{i=1}^{n}\lambda_i^2 \geq \sum_{i,j=1}^{n} a_{ij}^{(kl)}(x,t)\lambda_i\lambda_j \geq M_0 \sum_{i=1}^{n}\lambda_i^2 . \]

II. All coefficients of equations \((18.3_r)\), as well as the functions \(f_{kl}(x,t)\), \(f_{kl}^{(1)}(x)\), \(a_j^{(k)}(x,t)\), \(b_j^{(k)}(x,t)\), \(f_k^{(2)}(x,t)\), \(d_{jl}^{(k)}(x,t)\), \(h_{ji}^{(kl)}(x,t)\), \(f_{kl}^{(3)}(x,t)\), appearing in \((18.3_r)\)—\((18.6_r)\), are continuous in their domains of definition; moreover, for \(r=2\),
\(c_k^{(kl)}(x,t)\leq 0\), and for \(r=3\),
\(c_k^{(kl)}(x,t)\leq c<0\), \((x,t)\in \overline{D_T^{(l)}}\), \(k=1,2,\ldots,m\); \(c\) is a constant.

In addition,

\[ d_{kl}^{(k)}(x,t)\geq \delta>0,\qquad h_{kl}^{(kl)}(x,t)\geq 0, \]

\[ (x,t)\in \Gamma^{(1)},\quad k=1,2,\ldots,m;\quad l=1,2;\quad \delta \text{ is a constant}, \tag{18.8} \]

\[ a_k^{(k)}(x,t)+b_k^{(k)}(x,t)\geq a_0>0, \]

\[ a_k^{(k)}(x,t)\geq 0,\qquad b_k^{(k)}(x,t)\geq 0,\quad (x,t)\in \Gamma^{(2)}, \tag{18.9} \]

\[ k=1,2,\ldots,m;\quad a_0 \text{ is a constant}. \]

Condition (18.9) will sometimes be replaced by the stronger condition

\[ a_k^{(k)}(x,t)\geq a_0>0,\qquad b_k^{(k)}(x,t)\geq 0, \tag{18.10} \]

\[ (x,t)\in \Gamma^{(2)},\quad k=1,2,\ldots,m. \]

The parabolic systems \((18.3_r)\)—\((18.6_r)\) have a special form, since in the \((kl)\)-th equation \((18.3_r)\), for \(r=1,2,3\), the principal part contains only the function \(u_{kl}\) with the number of the equation. The lower-order terms of the \((kl)\)-th equation \((18.3_r)\), for \(r=1\), may contain the functions \(u_{ij}\) with arbitrary \(j=1,2,\ldots,m\), whereas for \(r=2\) the lower-order terms contain the functions \(u_{ij}\) only with indices \(j\leq k\). Similarly, the principal parts of the boundary conditions \((18.5_r)\) and of the conjugation conditions \((18.6_r)\), for \(r=1,2,3\), contain only the functions \(u_{kl}\), while the right-hand sides of these conditions may contain the functions \(u_{jl}\) with arbitrary indices \(j\) for \(r=1\), and with indices \(j\leq k-1\) for \(r=2\). A system of the form \((18.3_2)\)—\((18.6_2)\) (see \((18.7_2)\)) will be called a parabolic weakly coupled system.

Obviously, the model system (17.1)—(17.4) is a parabolic weakly coupled system representing a particular case of the system \((18.3_r)\)—\((18.6_r)\) (see \((18.7_r)\)) both for \(r=2\) and for \(r=3\).

If we put \(f_{kl}(x,t)\equiv 0\) in \((18.3_r)\), \(f_{kl}^{(1)}(x)\equiv 0\) in (18.4), \(f_k^{(2)}(x,t)\equiv 0\) in \((18.5_r)\), and \(f_{kl}^{(3)}(x,t)\equiv 0\) in \((18.6_r)\), then the resulting systems will be denoted by \((18.3_r')\), \((18.4_r')\), \((18.5_r')\), \((18.6_r')\). We shall also consider problems of the form \((18.3_r)\)—\((18.6_r)\) for a single parabolic equation with discontinuous coefficients

\[ L_l^{(r)}(u_l)=f_l(x,t),\quad (x,t)\in D_T^{(l)},\quad l=1,2, \tag{18.11\(_r\)} \]

\[ u_l(x,0)=f_l^{(1)}(x),\quad x\in \overline{\Omega}^{(l)},\quad l=1,2, \tag{18.12} \]

\[ l_1(u_2)\equiv l(u_2)=f^{(2)}(x,t),\quad (x,t)\in \Gamma^{(2)}, \tag{18.13} \]

\[ l_{1l}(u_1,u_2)\equiv l_l(u_1,u_2)=f_l^{(3)}(x,t),\quad (x,t)\in \Gamma^{(1)},\quad l=1,2. \tag{18.14} \]

\(((18.11_r), (18.12)—(18.14) are obtained from \((18.3_r)\)—\((18.6_r)\) for \(k=1\), with the index \(k=1\) omitted.)

We now introduce parabolic operators with constant coefficients

$$ L_{kl}^{(0)}(u_{kl}) \quad \text{and} \quad L_l^{(0)}(u_l), $$

where

$$ L_{kl}^{(0)}(u_{kl}) \equiv \sum_{i,j=1}^{n} a_{ij}^{(kl)} \frac{\partial^2 u_{kl}}{\partial x_i \partial x_j} -\frac{\partial u_{kl}}{\partial t}, \tag{18.15} $$

$$ k=1,2,\ldots,m;\quad l=1,2, $$

and \(L_l^{(0)}(u_l)\) is obtained from (18.15) by omitting the index \(k\). The matrix of constant coefficients \(a_{ij}^{(kl)}\) of the operator (18.15), \(\|a_{ij}^{(kl)}\|\), will be assumed positive definite and symmetric (with \(k\) and \(l\) regarded as fixed). We introduce the notation \(A_{ij}^{(kl)}\) for the elements of the matrix \(\|a_{ij}^{(kl)}\|^{1/2}\), \(a_{(kl)}^{ij}\) for the elements of the inverse matrix \(\|a_{ij}^{(kl)}\|^{-1}\), and \(A_{(kl)}^{ij}\) for the elements of the matrix \((\|a_{ij}^{(kl)}\|^{-1})^{1/2}\). We shall also consider the auxiliary problems \((18.3_0)\), (18.4), \((18.5_r)\), \((18.6_r)\) (for \(r=1,2,3\)), as well as the auxiliary problem for a single equation \((18.11_0)\), (18.12)—(18.14).

§ 19. UNIQUENESS OF THE SOLUTION OF PROBLEM \((18.3_2)\)—\((18.6_2)\)

Theorem 23. Suppose that, for problem \((18.3_2)\)—\((18.6_2)\) (see \((17.7_2)\)), conditions I, II, (18.8), (18.9) are satisfied. Suppose (see [12], § 3) that the surface \(\Gamma^{(l)}\) is such that for every point \(M(x,t)\in\Gamma^{(l)}\) \((0<t<T)\) there is a sphere \(S_M\) with center at the point \(\overline M(\overline x,\overline t)\in D_T^{(s)}\) (\(s=2\) for \(l=2\), or \(s=1,2\) for \(l=1\)), possessing the properties

$$ S_M \subset \overline D_T^{(s)}, \quad r(x,\overline x)\ne 0, $$

\(S_M\) has \(M(x,t)\) as its unique common point with \(\Gamma^{(l)}\), and, finally, if \(M_1(x_1,t_1)\in S_M\) with \(t_1<t\), then

$$ D^{(s)}(M_1)=\{\,\widetilde M(y,\tau),\ \widetilde M\in D^{(s)}(M)\ \text{for } \tau \le t_1\,\} $$

(here the symbol \(D^{(s)}(M_1)\) denotes the set of those points of \(D_T^{(s)}\) that can be joined to the point \(M_1\) by a curve lying entirely (except, possibly, for the point \(M_1\)) in the domain \(D_T^{(s)}\), and along which the coordinate \(\tau\) does not decrease). Then if \(u_{kl}^{(1)}(x,t)\) and \(u_{kl}^{(2)}(x,t)\) are two solutions of problem \((18.3_2)\)—\((18.6_2)\), continuous in \(\overline D_T^{(l)}\) and satisfying the same initial (18.4) and boundary \((18.5_2)\), \((18.6_2)\) conditions, then

$$ u_{kl}^{(1)}(x,t) \equiv u_{kl}^{(2)}(x,t),\quad (x,t)\in \overline D_T^{(l)},\quad k=1,2,\ldots,m;\quad l=1,2. $$

Proof. Put

$$ u_{kl}^{(1)}(x,t)-u_{kl}^{(2)}(x,t)=u_{kl}(x,t). $$

Obviously, \(u_{11}(x,t)\) satisfies the system \((18.3_2^0)\)—\((18.6_2^0)\) for \(k=1\). If \(u_{11}(x,t)\not\equiv 0\), put

$$ M_{1l}=\max_{\overline D_T^{(l)}} u_{1l}(x,t),\qquad m_{1l}=\min_{\overline D_T^{(l)}} u_{1l}(x,t). $$

Suppose that for at least one of the \(l\), \(M_{1l}>0\). Then 3 cases are possible: 1) \(M_{1l}>0\) for \(l=1,2\); 2) \(M_{11}>0\ge M_{12}\), and 3) \(M_{12}>0\ge M_{11}\). In case 1), by virtue of the maximum principle (see [12]), the function \(u_{1l}(x,t)\), continuous on \(\overline D_T^{(l)}\), can attain the value of its positive maximum \(M_{1l}\) only on the surface \(\Gamma^{(l)}\).

Let

\[ M_{1l}=u_{1l}(x_{1l},\,t_{1l}),\quad (x_{1l},\,t_{1l})\in \Gamma^{(1)}. \]

By the Vybornyi theorem ([12], Theorem 1),

\[ \frac{\partial u_{1l}(x_{1l},\,t_{1l})}{\partial \nu^{(1)}_{1}(x_{1l},\,t_{1l})}<0. \tag{19.1} \]

In view of (18.8), two possibilities are admissible:

a) \(\ h^{(1l)}_{1l}(x_{1l},\,t_{1l})=0\)

and

b) \(\ h^{(1l)}_{1l}(x_{1l},\,t_{1l})>0.\)

In case a), condition \((18.6_2)\) (for \(k=1\)), considered at the point \((x_{1l},\,t_{1l})\), contradicts (19.1). In case b), condition \((18.6_2)\) (for \(k=1\)), considered at the point \((x_{1l},\,t_{1l})\), gives, by virtue of (19.1), the inequalities

\[ u_{12}(x_{11},\,t_{11})-u_{11}(x_{11},\,t_{11})>0 \quad \text{for } l=1 \tag{19.2} \]

and

\[ u_{12}(x_{12},\,t_{12})-u_{11}(x_{12},\,t_{12})<0 \quad \text{for } l=2. \tag{19.3} \]

From (19.2) we have

\[ M_{12}\geq u_{12}(x_{11},\,t_{11})>u_{11}(x_{11},\,t_{11})=M_{11},\quad \text{i.e. } M_{12}>M_{11}, \tag{19.4} \]

while from (19.3)

\[ M_{12}=u_{12}(x_{12},\,t_{12})<u_{11}(x_{12},\,t_{12})\leq M_{11},\quad \text{i.e. } M_{12}<M_{11}. \]

The contradiction obtained shows the impossibility of case 1). In case 2), we see that if a) holds for \(l=1\), condition \((18.6_2^0)\) (for \(k=1,\ l=1\)) contradicts (19.1) for \(l=1\). If b) holds, then from (19.2) (see \((18.6_2^0)\) for \(k=l=1\)) we have (19.4), which contradicts the assumption \(M_{11}>0\geq M_{12}\). Thus case 2) (as well as case 3)) is impossible. Therefore

\[ u_{1l}(x,t)\leq 0,\quad (x,t)\in \overline{D}^{(l)}_{T}. \]

The supposition that, for at least one \(l\), \(m_{1l}<0\) similarly leads to a contradiction. Thus,

\[ u_{1l}(x,t)\equiv 0\quad (l=1,2)\quad \text{for } (x,t)\in \overline{D}^{(l)}_{T}. \tag{19.5} \]

Let us now consider the pair of functions \(u_{2l}(x,t)\) \((l=1,2)\), which, in view of the structure of the parabolic weakly coupled system, satisfies the system \((18.3_2^0)\)—\((18.6_2^0)\) for \(k=2\). We note that, by virtue of (19.5), in the system \((18.3_2^0)\)—\((18.6_2^0)\) for \(k=2\), only the functions \(u_{2l}(x,t)\) will enter, to which the arguments given above for \(u_{1l}(x,t)\) are fully applicable. Therefore, for \(k=2\) we have

\[ u_{kl}(x,t)\equiv 0,\quad (x,t)\in \overline{D}^{(l)}_{T},\quad l=1,2. \tag{19.6} \]

Repeating step by step the proof for the pairs \(u_{kl}(x,t)\) for \(k=3,4,\ldots,m\), using (19.6) for \(k-1\), we finally obtain (19.6) for \(k=1,2,\ldots,m\).

Remark. Theorem 23 on the uniqueness of the solution of problem \((18.3_2)\)—\((18.6_2)\) remains valid also when the condition

\[ c^{(kl)}_{k}(x,t)\leq 0,\quad (x,t)\in \overline{D}^{(l)}_{T},\quad k=1,2,\ldots,m;\quad l=1,2 \]

in II by the more general condition

\[ c_k^{(kl)}(x,t)\leq c<+\infty,\quad (x,t)\in \overline{D_T^{(l)}};\quad k=1,2,\ldots,m;\ l=1,2; \tag{19.7} \]

\(c\) is a constant.

Indeed, for \(c>0\) it suffices to make the usual substitution

\[ u_{kl}(x,t)=\bar u_{kl}(x,t)e^{-\lambda t} \]

with \(\lambda>c\) (\(\lambda\) a constant) and to observe that

\[ \cos\bigl(\nu_k^{(sl)}(x,t),Ot\bigr)=0,\quad (x,t)\in \Gamma_t^{(s)};\quad k=1,2,\ldots,m; \]

\[ l=1,2\ \text{for } s=1 \quad \text{and} \quad l=2\ \text{for } s=2. \]

Then \(\bar u_{kl}(x,t)\) is a solution of a system of the form \((18.3_2)\)—\((18.6_2)\), for which all the conditions of Theorem 23 are satisfied.

§ 20. On the maximum principle for the parabolic weakly coupled system \((18.3_3)\)—\((18.6_3)\) with discontinuous coefficients

In § 19 it was shown that for parabolic weakly coupled systems \((18.3_2)\)—\((18.6_2)\) a uniqueness theorem holds. We shall now consider a narrower class of systems of the form \((18.3_3)\)—\((18.6_3)\) (see \((18.7_3)\)). This class, as it turns out, possesses a number of interesting properties following from the maximum principle and making it possible to give an a priori estimate for \(\left.|u_{kl}|\right|_0^{D_T^{(l)}}\). We note that analogous properties were previously discovered and studied in [13], [14] by the author jointly with V. N. Maslennikova for parabolic equations with discontinuous coefficients when solving boundary value problems in a somewhat different formulation. In what follows in § 20 both the methods and the constructions of the work cited [14] will be used. We also state the conditions used in Lemmas 22–29.

III. The coefficients of equations \((18.3_3)\)—\((18.11_3)\) belong to the class

\[ H^{0,\alpha,\frac{\alpha}{2}}\bigl(\overline{D_T^{(l)}}\bigr),\qquad (0<\alpha<1). \]

IV. The surfaces \(\Gamma^{(l)}\) are of type \(Л^{1,\alpha;\alpha/2}_{1,1,(1+\alpha)/2}\), \((0<\alpha<1)\).

Introduce two functions \(v_l(x,t)\) \((l=1,2)\), satisfying equations \((18.11_3^0)\) with initial conditions

\[ v_l(x,0)=A_d^{(l)}(x),\quad x\in \overline{\Omega^{(l)}};\quad l=1,2 \tag{20.1} \]

and with boundary conditions

\[ v_l(x,t)=A>0,\quad (x,t)\in \Gamma^{(s)};\quad A=\text{const} \]

\[ (s=1\ \text{for } l=1 \ \text{and}\ s=1,2\ \text{for } l=2), \tag{20.2} \]

where \(A_d^{(l)}(x)\) are functions continuous in \(\overline{\Omega^{(l)}}\) together with their derivatives of sufficiently high order, and, moreover (cf. [14], p. 385):

\[ A_d^{(l)}(x)= \begin{cases} A, & \text{for } x\in \Gamma_0^{(s)}=\Gamma^{(s)}\cap\{t=0\} \\ & (s=1\ \text{for } l=1 \ \text{and}\ s=1,2\ \text{for } l=2),\\[4pt] 0, & \text{for } x \text{ at a distance greater than } d \text{ from } \Gamma_0^{(s)} \text{ from } (18.1). \end{cases} \]

Lemma 22 (see Lemma 1 [14]). Suppose that conditions I—III (for \((18.11_3^0)\)) and IV are satisfied. Suppose that the functions \(A_d^{(l)}(x)\) from (20.1) belong to the class
\(H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\bigl(\overline{\Omega}^{(l)}\bigr)\) \((0<\alpha\leqslant 1)\). Then, for the solution \(v_l(x,t)\) of problem \((18.11_3^0)\), (20.1), (20.2), the inequalities

\[ \begin{cases} 0 \leqslant v_l(x,t)\leqslant A, & (x,t)\in \overline{D_T^{(l)}},\\ 0 \leqslant v_l(x,t)< A, & (x,t)\in D_T^{(l)}, \end{cases} \tag{20.3} \]

\[ \frac{\partial v_l(x,t)}{\partial \nu^{(sl)}(x,t)}\leqslant -r \quad \text{for } (x,t)\in \Gamma^{(s)}, \tag{20.4} \]

hold, where \(l=1,2\) for \(s=1\) and \(l=2\) for \(s=2\); here \(r>0\) is a constant and

\[ l_1(v_2)<0 \quad \text{on } \Gamma^{(2)}. \tag{20.5} \]

Lemma 23. Suppose that, for problem \((18.11_3^0)\), \((18.12^\circ)\), \((18.13^\circ)\), (18.14), conditions I—IV and (18.8), (18.9) are satisfied. Then, for the solution \(u_l(x,t)\) of this problem, continuous on \(\overline{D_T^{(l)}}\), the estimate

\[ \left|u_l\right|_{0}^{D_T^{(l)}} \leqslant (r\delta)^{-1} A \max_{i=1,\,2} \left|f_i^{(3)}\right|_{0}^{\Gamma^{(1)}} . \tag{20.6} \]

holds.

Proof. Following Theorem 1 [14], consider the auxiliary functions

\[ w_l(x,t)=(r\delta)^{-1}F^{(3)}v_l(x,t)-u_l(x,t), \qquad (x,t)\in D_T^{(l)}, \tag{20.7} \]

where

\[ F^{(3)}=\max_{l=1,\,2}\left|f_l^{(3)}\right|_{0}^{\Gamma^{(1)}}. \]

Obviously,

\[ L_l^{(3)}(w_l)=0,\qquad l_1(w_2)<0, \]

\[ w_l(x,0)=(r\delta)^{-1}F^{(3)}A_d^{(l)}(x)\geqslant 0, \qquad l=1,2. \tag{20.8} \]

We shall prove that

\[ w_l(x,t)\geqslant 0 \quad \text{for } (x,t)\in \overline{D_T^{(l)}},\qquad l=1,2. \tag{20.9} \]

Assuming the contrary, we arrive at three possibilities:

1) \(w_l(x,t)\), for \(l=1,2\), attains the value of its negative minimum (on \(\overline{D_T^{(l)}}\)) at a point \((x_l,t_l)\in \overline{D_T^{(l)}}\), \((l=1,2)\);

2) \(w_1(x,t)\geqslant 0\) for \((x,t)\in \overline{D_T^{(1)}}\), but \(w_2(x,t)\) attains the value of its negative minimum (on \(\overline{D_T^{(2)}}\)) at a point \((x_2,t_2)\);

3) \(w_2(x,t)\geqslant 0\) for \((x,t)\in \overline{D_T^{(2)}}\), but \(w_1(x,t)\) attains the value of its negative minimum (on \(\overline{D_T^{(1)}}\)) at a point \((x_1,t_1)\).

In case 1), by the maximum principle, it follows from (20.8) that the function \(w_l(x,t)\), continuous on \(\overline{D_T^{(l)}}\), \((l=1,2)\), can attain the value of its negative minimum (on \(\overline{D_T^{(l)}}\)) only at a point \((x_l,t_l)\in \Gamma^{(1)}\) with \(t_l>0\).

It follows from (20.2) and (20.3) that at the point \((x_l,t_l)\) the function \(u_l(x,t)\) attains the value of its positive maximum (on \(\overline{D}_{T}^{(l)}\)), i.e.

\[ u_l(x_l,t_l)=M_l=\max_{\overline{D}_{T}^{(l)}} u_l(x,t)>0. \tag{20.10} \]

But then, by Vyborny’s theorem ([12], Theorem 1),

\[ \frac{\partial u_l(x_l,t_l)} {\partial \nu^{(ll)}(x_l,t_l)}<0, \tag{20.11} \]

\[ \frac{\partial w_l(x_l,t_l)} {\partial \nu^{(ll)}(x_l,t_l)}>0 \quad \text{for } (x_l,t_l)\in \Gamma^{(1)},\quad l=1,2. \tag{20.12} \]

Two cases are possible: 1.1) \(M_1>M_2\) and 1.2) \(M_1\leqslant M_2\).
In case 1.1) we have

\[ u_2(x_1,t_1)-M_1<0. \tag{20.13} \]

From (18.14) with \(l=1\) we obtain

\[ -d_1(x_1,t_1)\, \frac{\partial u_1(x_1,t_1)} {\partial \nu^{(11)}(x_1,t_1)} +h_1^{(1)}(x_1,t_1)\bigl(M_1-u_2(x_1,t_1)\bigr) = \]

\[ = f_1^{(3)}(x_1,t_1). \tag{20.14} \]

In view of (20.11) for \(l=1\) and (20.13), both terms on the left in (20.14) are nonnegative; therefore for \(l=1\)

\[ \left| d_l(x_l,t_l)\, \frac{\partial u_l(x_l,t_l)} {\partial \nu^{(ll)}(x_l,t_l)} \right| \leqslant \left|f_l^{(3)}(x_l,t_l)\right| \leqslant F^{(3)}. \tag{20.15} \]

On the other hand, from (20.12) (for \(l=1\)) it follows, for \(l=1\), that

\[ (r\delta)^{-1}F^{(3)} \frac{\partial v_l(x_l,t_l)} {\partial \nu^{(ll)}(x_l,t_l)} > \frac{\partial u_l(x_l,t_l)} {\partial \nu^{(ll)}(x_l,t_l)} \tag{20.16} \]

and, in view of (20.4), (18.8), we obtain for \(l=1\)

\[ \left| \frac{\partial u_l(x_l,t_l)} {\partial \nu^{(ll)}(x_l,t_l)} \right| > \delta^{-1}F^{(3)}, \]

and therefore, for \(l=1\),

\[ \left| d_l(x_l,t_l)\, \frac{\partial u_l(x_l,t_l)} {\partial \nu^{(ll)}(x_l,t_l)} \right| > F^{(3)}, \tag{20.17} \]

which contradicts (20.15) (for \(l=1\)).

In case 1.2) we have

\[ u_1(x_2,t_2)-M_2\leqslant 0. \tag{20.18} \]

From (18.14) with \(l=2\) we obtain

\[ d_2(x_2,t_2)\, \frac{\partial u_2(x_2,t_2)} {\partial \nu^{(12)}(x_2,t_2)} + \]

\[ +\,h_2^{(2)}(x_2,t_2)\bigl(u_1(x_2,t_2)-M_2\bigr) = f_2^{(3)}(x_2,t_2), \tag{20.19} \]

where both terms on the left in (20.19), in view of (20.18), (18.8), and (20.11) (for \(l=2\)), are nonpositive, which entails (20.15) (for \(l=2\)). From (20.12) (for \(l=2\)) it fol-

gives (20.17) (for \(l=2\)) and, in view of (20.4), (18.8), we obtain (20.17) (for \(l=2\)), which contradicts (20.15) (for \(l=2\)).

In case 2), by virtue of (20.3), (20.7), we have

\[ u_1(x,t)\leqslant (r\delta)^{-1}F^{(3)}A \quad \text{for } (x,t)\in \overline{D_T^{(1)}} . \tag{20.20} \]

By the maximum principle, from (20.8) (for \(l=2\)) it follows that \((x_2,t_2)\in \Gamma^{(1)}\) (with \(t_2>0\)), and moreover, by virtue of (20.2), (20.3), (20.10)—(20.12) hold (for \(l=2\)). Two cases are possible:

\[ \text{2.1) }\quad M_2\leqslant (r\delta)^{-1}F^{(3)}A \]

and

\[ \text{2.2) }\quad M_2>(r\delta)^{-1}F^{(3)}A. \]

In case 2.1) we have

\[ u_2(x,t)\leqslant (r\delta)^{-1}F^{(3)}A \quad \text{for } (x,t)\in \overline{D_T^{(2)}}, \]

whence, in view of (20.20), the validity of the estimate (20.6) from one side follows. In case 2.2), as in case 1.2), we arrive at a contradiction. Case 3) is considered analogously to case 2). Thus estimate (20.9) is proved. From (20.9), in view of (20.7), (20.3), we obtain that the estimate (20.6) is valid from above. The proof of the estimate (20.6) from below is carried out in a completely analogous way, if one considers the auxiliary function

\[ w_l^*(x,t)=(r\delta)^{-1}F^{(3)}v_l(x,t)-u_l(x,t), \]

\[ l=1,2;\quad (x,t)\in D_T^{(l)}. \]

Lemma 2⁴. Suppose that for problem (18.11₃), (18.12)—(18.14) conditions I—IV and (18.8), (18.9) are fulfilled. Suppose, moreover, that

\[ f_l(x,t)\geqslant 0 \quad (f_l(x,t)\leqslant 0),\quad (x,t)\in \overline{D_T^{(l)}},\quad l_i=1,2, \tag{20.21} \]

\[ f^{(2)}(x,t)\geqslant 0 \quad (f^{(2)}(x,t)\leqslant 0),\quad (x,t)\in \Gamma^{(2)}, \tag{20.22} \]

\[ (-1)^l f_l^{(2)}(x,t)\leqslant 0 \quad \left((-1)^l f_l^{(3)}(x,t)\geqslant 0\right), \tag{20.23} \]

\[ (x,t)\in \Gamma^{(1)},\quad l=1,2. \]

Then the solution \(u_l(x,t)\), continuous on \(\overline{D_T^{(l)}}\), of problem (18.11₃)—(18.14) cannot attain the value of its positive maximum (negative minimum) on \(\overline{D_T^{(l)}}\) on any one of the boundary surfaces \(\Gamma^{(l)}\) \((l=1,2)\).

Proof. By the maximum principle and the theorem of Vyborny ([12], Theorem 1), it follows from (20.21), (20.22) that the function \(u_l(x,t)\), continuous on \(\overline{D_T^{(l)}}\), can attain the value of its positive maximum either at \(t=0\), or on \(\Gamma^{(1)}\). We shall show that the second assumption is false.

Assuming the contrary and using the notation (20.10), we shall distinguish the following cases:

\[ \text{1) }\quad M_l>0,\quad l=1,2, \]

\[ \text{2) }\quad M_1>0,\quad u_2(x,t)\leqslant 0 \quad \text{for } (x,t)\in \overline{D_T^{(2)}}, \]

\[ \text{3) }\quad M_2>0,\quad u_1(x,t)\leqslant 0 \quad \text{for } (x,t)\in \overline{D_T^{(1)}}. \]

In case 1) we have two possibilities: 1.1) \(M_1>M_2\) and 1.2) \(M_1\leqslant M_2\).

In case 1.1) consider (20.14). The left-hand side of (20.14), in view of (20.13) and (20.11) (for \(l=1\)), is positive, which contradicts (20.23) (for \(l=1\)). In case 1.2) the left-hand side of (20.19), in view of (20.18) and (20.11) (\(l=2\)), is negative, which contradicts (20.23) for \(l=2\).

Case 2), in view of (20.13), reduces to the case 1.1) considered, and case 3), in view of (20.18), to 1.2).

Thus the value of a positive maximum on \(\overline{D}^{(l)}_T\) for the function \(u_l(x,t)\) cannot be attained on \(\Gamma^{(1)}\). In the case of a negative minimum the proof is analogous.

Lemma 25. If, under the fulfillment of all the conditions of Lemma 24, the inequalities

\[ f_l^{(1)}(x) \ge 0 \qquad \bigl(f_l^{(1)}(x) \le 0\bigr) \quad \text{for } x \in \overline{\Omega}^{(l)},\quad l=1,2, \]

hold, then

\[ u_l(x,t) \ge 0 \qquad \bigl(u_l(x,t) \le 0\bigr) \quad \text{for } (x,t)\in \overline{D}^{(l)}_T,\quad l=1,2. \]

Lemma 25 follows from Lemma 24.

Lemma 26. Suppose that for the problem (18.11\(_3\)), (18.12°), (18.13°), (18.14) conditions I–IV, (18.8), (18.9) are fulfilled, and, moreover, for (18.11\(_3\))

\[ c_1^{(l)}(x,t)<0,\quad (x,t)\in \overline{D}^{(l)}_T,\quad l=1,2, \tag{20.24} \]

then for the solution of the problem (18.11\(_3\)), (18.12°), (18.13°), (18.14), continuous on \(\overline{D}^{(l)}_T\), the estimate holds:

\[ |u_l|^{D^{(l)}_T}_0 \le m+(r\delta)^{-1}A \max_{i=1,2}|f_i^{(3)}|^{\Gamma^{(1)}}_0, \]

where

\[ m=\max_{i=1,2} \left\{ \left[\min_{\overline{D}^{(i)}_T}|c_1^{(i)}(x,t)|\right]^{-1} |f_i|^{D^{(i)}_T}_0 \right\}. \tag{20.25} \]

The proof of Lemma 26 is completely analogous to the proof of Lemma 23 if, following Theorem 3 [14], instead of (20.7) one considers the auxiliary function (see (20.25))

\[ w_i(x,t)=(r\delta)^{-1}F^{(3)}v_l(x,t)+m-u_l(x,t) \]

and notes that, instead of (20.8), we have

\[ L_l^{(3)}(w_l)=c_1^{(l)}(x,t)m+f_l(x,t)\le 0, \quad (x,t)\in D^{(l)}_T, \]

\[ l_1(w_2)=-mb_1(x,t)+(r\delta)^{-1}F^{(3)}l_1(v_2)<0; \quad (x,t)\in \Gamma^{(2)}, \]

\[ w_l(x,0)=m+(r\delta)^{-1}F^{(3)}A_d^{(l)}(x)>0, \quad x\in \overline{\Omega}^{(l)},\quad l=1,2. \]

Lemma 27. Suppose that for the problem (18.11\(_3\)), (18.12°), (18.13), (18.14) conditions I–IV, (18.8), (18.9), (20.24) are fulfilled. Suppose that, for any functions \(\psi_l^{(1)}(x)\), \(\psi^{(2)}(x,t)\), and \(\psi_l^{(3)}(x,t)\) from the classes \(C^{(s_1)}(\overline{\Omega}^{(l)})\), \(C^{(s_2)}(\Gamma^{(2)})\), and \(C^{(s_3)}(\Gamma^{(1)})\), \(s_i\ge 0\), respectively, there exists a solution of the problem (18.11\(_3\)), (18.12) (with \(f_l^{(1)}(x)=\psi_l^{(1)}(x)\)), (18.13) (with \(f^{(2)}(x,t)=\psi^{(2)}(x,t)\)), (18.14) (with \(f_l^{(3)}(x,t)=\psi_l^{(3)}(x,t)\)) continuous on \(\overline{D}^{(l)}_T\) (where \(\psi_l^{(1)}(x)\), \(\psi^{(2)}(x,t)\), and \(\psi_l^{(3)}(x,t)\) are compatible on the edges \(\Gamma^{(s)}\cap \overline{\Omega}^{(l)}\) (\(l=1,2\) for \(s=1\) and \(l=2\) for \(s=2\)) by virtue of the conditions

problem). Suppose moreover that \(a_1(x,t)\), \(b_1(x,t)\), and \(\nu^{(22)}(x,t)\) from (18.13) belong to \(C^{(s_2)}(\Gamma^{(2)})\); \(d_i(x,t)\), \(h_i^{(1)}(x,t)\) from (18.14) belong to the class \(C^{(s_3)}(\Gamma^{(1)})\). Then the solution \(u_1(x,t)\) of problem \((18.11_3)\), \((18.12^0)\), (18.13), (18.14), continuous in \(\overline{D}_T^{(l)}\), satisfies the estimate (see (20.25))

\[ |u_l|^{D_T^{(l)}}_0 \leq M\bigl(|f_l|^{D_T^{(l)}}_0,\ |f^{(2)}|^{\Gamma^{(2)}}_0,\ |f_i^{(3)}|^{\Gamma^{(1)}}_0\bigr) \equiv m+ \]

\[ + (r\delta)^{-1} A \max_{i=1,\,2} |f_i^{(3)}|^{\Gamma^{(1)}}_0+ \]

\[ + \max\left(1,\ \left[\min_{\Gamma^{(2)}}\{a_1(x,t)+b_1(x,t)\}\right]^{-1}|f^{(2)}|^{\Gamma^{(2)}}_0\right)\times \]

\[ \times\left[1+r^{-1}nAK_2+K_2\max_{s=1,\,2}\{m_s^0\cdot m_s^1\}\right], \tag{20.26} \]

where

\[ m_s^0=\left[\min_{\overline{D}_T^{(s)}} |c_1^{(s)}(x,t)|\right]^{-1},\quad m_s^1= \]

\[ =\sum_{i,j=1}^{n}|a_{ij}^{(s)}|^{D_T^{(s)}}_0+ \sum_{i=1}^{n}|b_i^{(s)}|^{D_T^{(s)}}_0+ |c_1^{(s)}|^{D_T^{(s)}}_0+1, \]

where the constants \(K_i>0\) \((i=1,2)\) depend neither on the coefficients of equation \((18.11_3)\) nor on the functions \(f_l(x,t)\), \(f_{(x,t)}^{(2)}\), and \(f_i^{(3)}(x,t)\).

Proof. Represent the solution \(u_l(x,t)\) of problem \((18.11_3)\), \((18.12^0)\), (18.13), (18.14) in the form

\[ u_l(x,t)=u_l^{(1)}(x,t)+u_l^{(2)}(x,t), \tag{20.27} \]

where \(u_l^{(1)}(x,t)\) is the solution of problem \((18.11_3)\), \((18.12^0)\), (18.13), \((18.14^0)\), and \(u_l^{(2)}(x,t)\) is the solution of problem \((18.11_3)\), \((18.12^0)\), \((18.13^0)\), and (18.14). By Lemma 26, for \(u_l^{(2)}(x,t)\) we have estimate (20.25). We shall estimate \(|u_l^{(1)}|^{D_T^{(l)}}_0\), following Theorem 4 of [14]. Introduce a smooth function \(U(x,t)\), defined in the domain \(\overline{D}_T\), for which \(\dfrac{\partial U(x,t)}{\partial \nu^{(22)}(x,t)}\) for \((x,t)\in\Gamma^{(2)}\) belongs to the class \(C^{(s_2)}(\Gamma^{(2)})\), and moreover

\[ -1\leq U(x,t)<0 \quad \text{for } (x,t)\in\overline{D}_T \tag{20.28} \]

and

\[ \frac{\partial U(x,t)}{\partial \nu^{(22)}(x,t)}<0 \quad \text{for } (x,t)\in\Gamma^{(2)}. \tag{20.29} \]

Then, by (20.29), (18.9), and the conditions of Lemma 27, we have

\[ l_1(U)\equiv \psi^{(2)}(x,t)>0,\quad (x,t)\in\Gamma^{(2)}, \tag{20.30} \]

where

\[ \psi^{(2)}(x,t)\in C^{(s_2)}(\Gamma^{(2)}). \]

Choose a constant \(K_1>0\), depending only on \(U(x,t)\), so that

\[ \min_{\Gamma^{(2)}}\psi^{(2)}(x,t)\geq K_1^{-1}\min_{\Gamma^{(2)}}(a_1(x,t)+b_1(x,t)). \tag{20.31} \]

Moreover, choose a constant \(K_2>0\) such that, for \((x,t)\in \overline{D}_T\), the estimate

\[ |U|_{\overline{D}_T}\leq K_2 . \tag{20.32} \]

holds.

Let \(\overline{u}_l(x,t)\) be a solution, continuous on \(\overline{D}_T^{(l)}\), of problem \((18.11_3^0)\), \((18.12)\) (with \(f_l^{(1)}(x)\equiv U(x,0)\)), \((18.13)\) (with \(f^{(2)}(x,t)\equiv \psi^{(2)}(x,t)\) from (20.30)) and \((18.14)\) (with \(f_l^{(3)}(x,t)\equiv (-1)^l d_l(x,t)\dfrac{\partial U(x,t)}{\partial \nu^{(1l)}(x,t)}\)). The existence of such a solution is postulated in the hypotheses of Lemma 27 and will follow from Theorem 29 of §22.

Introduce the auxiliary functions

\[ w_l(x,t)=m_l\overline{u}_l(x,t)-u_l^{(1)}(x,t)-m_l r^{-1}nK_2v_l(x,t),\qquad m_1=1, \]

where \(v_l(x,t)\) are taken from Lemma 22, and

\[ m_2=\bigl[\min_{\Gamma^{(2)}}|\psi^{(2)}(x,t)|\bigr]^{-1}|f^{(2)}|_{\Gamma_0^{(2)}}>0 . \tag{20.33} \]

By virtue of (20.28)—(20.32), the definitions of \(u_l^{(1)}(x,t)\), \(\overline{u}_l(x,t)\), and (20.1)—(20.5), we see that \(w_l(x,t)\) satisfies equation \((18.11_3^0)\) with the additional conditions

\[ w_l(x,0)=m_l\bigl(U(x,0)-r^{-1}nK_2A_d^{(l)}(x)\bigr)<0,\qquad x\in \overline{\Omega}^{(l)},\quad l=1,2, \]

\[ l_1(w_2)=m_2\psi^{(2)}(x,t)-f^{(2)}(x,t)-m_2r^{-1}nK_2l_1(v_2)\geq 0 \]

by virtue of (20.5), (20.33),

\[ (-1)^s l_s(w_1,w_2)\equiv m_s d_s(x,t)\left[ \frac{\partial U(x,t)}{\partial \nu^{(1s)}(x,t)} - r^{-1}nK_2 \frac{\partial v_s(x,t)}{\partial \nu^{(1s)}(x,t)} \right]>0,\qquad s=1,2 \]

by virtue of (20.2), (20.4), (20.32). Application of Lemma 25 gives

\[ w_l(x,t)\leq 0,\qquad (x,t)\in \overline{D}_T^{(l)},\quad l=1,2. \tag{20.34} \]

Consider the auxiliary functions

\[ \widehat{w}_l(x,t)=m_l\overline{u}_l(x,t)+u_l^{(1)}(x,t)- \]

\[ -r^{-1}nK_2v_l(x,t),\qquad (x,t)\in D_T^{(l)},\quad l=1,2. \]

Obviously, \(\widehat{w}_l(x,t)\) satisfies equation \((18.11_3^0)\) with the additional conditions

\[ \widehat{w}_l(x,0)<0,\qquad x\in \overline{\Omega}^{(l)};\quad l_1(\widehat{w}_2)\geq 0, \]

\[ (-1)^s l_s(\widehat{w}_1,\widehat{w}_2)>0,\qquad s=1,2. \]

Therefore, by Lemma 25, we have

\[ \widehat{w}_l(x,t)\leq 0,\qquad (x,t)\in \overline{D}_T^{(l)},\quad l=1,2. \tag{20.35} \]

From (20.34), (20.35), and (20.3) we have the estimates

\[ \left|u_l^{(1)}(x,t)\right| \leq m_l\left(\left|\bar u_l(x,t)\right|+r^{-1}nK_2A\right), \]

\[ (x,t)\in \bar D_T^{(l)},\quad l=1,2. \tag{20.36} \]

To estimate \(\bar u_l(x,t)\), consider the functions

\[ \overline{\overline u}_l(x,t)=\bar u_l(x,t)-U(x,t). \tag{20.37} \]

Obviously, \(\overline{\overline u}_l(x,t)\) is a solution of problem \((18.11_3)\) (with \(f_l(x,t)\equiv -L_l^{(3)}(U)\)), \((18.12^0)\)—\((18.14^0)\). Hence, by Lemma 26, we have the estimate (see (20.26))

\[ \left|\overline{\overline u}_l\right|_0^{D_T^{(l)}} \leq \max_{i=1,2}\left[m_i^0\left|L_i^{(3)}(U)\right|_0^{D_T^{(i)}}\right] \leq K_2\max_{s=1,2}\left|m_s^0\cdot m_s^1\right|. \tag{20.38} \]

From the representations (20.27), (20.37) and the estimates (20.25), (20.36), (20.28), and (20.38) we obtain the estimate (20.26) of Lemma 27.

Lemma 28. If \(u_l(x,t)\) is a solution, continuous on \(\bar D_T^{(l)}\), of problem \((18.11_3^0)\), \((18.12)\), \((18.13^0)\), \((18.14^0)\), then the estimate holds

\[ \left|u_l\right|_0^{D_T^{(l)}}\leq \left|f_l^{(1)}\right|_0^{\Omega^{(l)}},\quad l=1,2. \tag{20.39} \]

Proof. By Lemma 24, the function \(u_l(x,t)\) can attain the value of its positive maximum or negative minimum on \(\bar D_T^{(l)}\) only when \(t=0\), whence the estimate (20.39) follows.

Lemma 29. If the hypotheses of Lemma 27 are satisfied and, for the functions \(f_l^{(1)}(x)\), the problem \((18.11_3^0)\), \((18.12)\), \((18.13^0)\), \((18.14^0)\) is solvable, then the solution \(u_l(x,t)\) of problem \((18.11_3)\)—\((18.14)\), continuous on \(\bar D_T^{(l)}\), satisfies on \(\bar D_T^{(l)}\) the inequality

\[ \left|u_l\right|_0^{D_T^{(l)}} \leq \left|f_l^{(1)}\right|_0^{\Omega^{(l)}} + M\left( \left|f_l\right|_0^{D_T^{(l)}}, \left|f^{(2)}\right|_0^{\Gamma^{(2)}}, \left|f_l^{(3)}\right|_0^{\Gamma^{(1)}} \right), \tag{20.40} \]

where the constant \(M(\ldots)\) is taken from (20.26).

Proof. Represent the solution of problem \((18.11_3)\)—\((18.14)\) \(u_l(x,t)\) in the form

\[ u_l(x,t)=u_l^{(1)}(x,t)+u_l^{(2)}(x,t),\quad l=1,2, \]

where \(u_l^{(s)}(x,t)\) \((s=1,2)\) are solutions of the problems \((18.11_3)\), \((18.12^0)\), \((18.13)\), \((18.14)\) and \((18.11_3^0)\), \((18.12)\), \((18.13^0)\), \((18.14^0)\), respectively, continuous on \(\bar D_T^{(l)}\). By Lemmas 27 and 28, applied to \(u_l^{(s)}(x,t)\), \(s=1,2\), we obtain the estimate (20.40) of Lemma 29.

Theorem 24. Suppose that for problem \((18.3_3)\)—\((18.6_3)\) (see (18.7)) conditions I, II (for \(r=3\)) of §18, and III, IV of §20, as well as (18.8), (18.9), are satisfied. Suppose, moreover, that

\[ \left|a_{ij}^{(kl)}\right|_0^{D_T^{(l)}}\leq A^{(k)},\quad \left|b_{kj}^{(kl)}\right|_0^{D_T^{(l)}}\leq B^{(k)}, \]

\[ \left|c_i^{(kl)}\right|_0^{D_T^{(l)}}\leq C^{(k)},\quad \left|f_{kl}\right|_0^{D_T^{(l)}}\leq M_k, \]

\[ \left|f_{kl}^{(1)}\right|_0^{\Omega^{(l)}}\leq M_k^{(1)},\quad \left|f_k^{(2)}\right|_0^{\Gamma^{(2)}}\leq M_k^{(2)}, \]

\[ \left|f_{kl}^{(3)}\right|_0^{\Gamma^{(1)}}\leq M_k^{(3)},\quad \left|b_i^{(k)}\right|_0^{\Gamma^{(2)}}\leq b_k,\quad \left|h_{ij}^{(kl)}\right|_0^{\Gamma^{(1)}}\leq h_k. \]

V. Let, for arbitrary

\[ \psi_{kl}^{(1)}(x)\in C^{(s_1)}\left(\overline{\Omega}^{(l)}\right),\qquad \psi_k^{(2)}(x,t)\in C^{(s_2)}\left(\Gamma^{(2)}\right), \]

\[ \psi_{kl}^{(3)}(x,t)\in C^{(s_3)}\left(\Gamma^{(1)}\right)\qquad (s_i\geq 0) \]

and for any fixed \(k\) \((k=1,2,\ldots,m)\), there exist a solution \(\bar u_{kl}(x,t)\), continuous on \(\overline D_T^{(l)}\), of problem \((18.3_3^0)\) (with \(c_i^{(kl)}(x,t)\equiv 0\) for \(1\leq i\leq k-1\)), \((18.4)\) (with \(f_{kl}^{(1)}(x)\equiv\psi_{kl}^{(1)}(x)\)), \((18.5_3)\) (with \(f_k^{(2)}(x,t)\equiv\psi_k^{(2)}(x,t)\)) and \((18.6_3)\) (with \(f_{kl}^{(3)}(x,t)\equiv\psi_{kl}^{(3)}(x,t)\)); moreover it is assumed that the functions \(a_k^{(k)}(x,t)\), \(b_k^{(22)}(x,t)\), \(v_k^{(22)}(x,t)\) from \((18.5_3)\) and \(d_{kl}^{(k)}(x,t)\), \(h_{kl}^{(kl)}(x,t)\), \(v_k^{(11)}(x,t)\) from \((18.6_3)\) belong to the classes \(C^{(s_2)}(\Gamma^{(2)})\) and \(C^{(s_3)}(\Gamma^{(1)})\), respectively.

VI. Finally, let problem \((18.3_3^0)\), \((18.4)\), \((18.5_3^0)\), \((18.6_3^0)\) be solvable.

Then, for the solution \(u_{kl}(x,t)\), continuous on \(\overline D_T^{(l)}\) \((l=1,2;\ k=1,2,\ldots,m)\), of problem \((18.3_3)\)—\((18.6_3)\), the estimate

\[ \left|u_{kl}\right|_{0}^{D_T^{(l)}}\leq M\left(r,d,a_0,b_j,h_j,A,c,A^{(i)},B^{(j)},C^{(j)},M_j^{(i)}; \right. \]

\[ \left. j=1,2,\ldots,k;\ i=1,2,3 \right). \tag{20.41} \]

holds.

Proof. Setting \(k=1\) in system \((18.3_3)\)—\((18.6_3)\), by virtue of \((20.40)\) of Lemma 29 we obtain estimate \((20.41)\) for \(k=1\). For \(k\geq 2\), we transfer to the right-hand side of \((18.3_3)\) the terms

\[ \sum_{i=1}^{k-1} c_i^{(kl)}(x,t)\,u_{il}(x,t), \]

denoting the resulting equation by \((18.3_3)^*\). For \(k\geq 2\), \(u_{kl}(x,t)\) may be regarded as a solution of problem \((18.3_3)^*\), \((18.4)\)—\((18.6_3)\) for fixed \(k\), where the right-hand sides of equation \((18.3_3)^*\) and of the boundary conditions \((18.5_3)\), \((18.6_3)\) depend linearly on \(u_{il}(x,t)\) with \(i=1,2,\ldots,k-1\). Repeatedly applying Lemma 29, we obtain estimate \((20.41)\) for \(k=2\), then for \(k=3\), and so on. Thus, estimate \((20.41)\) holds for any \(k=1,2,\ldots,m\).

Remark 1. From Theorem 29 of § 22 it follows that conditions V, VI are certainly satisfied if

\[ s_1=2+\alpha,\qquad C^{(s_2)}(\Gamma^{(2)})=H_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}(\Gamma^{(2)}), \]

\[ C^{(s_3)}(\Gamma^{(1)})=H_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}(\Gamma^{(1)}) \]

and \((18.10)\) holds.

Remark 2. Theorem 24 on the a priori estimate of the modulus of the solution of problem \((18.3_3)\)—\((18.6_3)\) remains valid if condition

\[ c_k^{(kl)}(x,t)\leq c<0 \]

in II is replaced by the more general condition \((19.7)\) (cf. the remark to Theorem 23 of § 19).

(To be continued)

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Received by the editors
January 24, 1964

Moscow State University
named after M. V. Lomonosov