UDC 517.946.9

MATHEMATICS

L. I. KAMYNIN

ON A PROBLEM OF BIOPHYSICS

(Presented by Academician S. L. Sobolev, 1 XI 1965)

The investigation of the question of the distribution of concentrations of substances participating in the vital processes of a living cell leads (see ((^1))) to the problem for a system of diffusion equations

[
\begin{gathered}
a_{11}\Delta u_{11}-(c_2+c_3)u_{11}-\partial u_{11}/\partial t=0,\qquad
a_{21}\Delta u_{21}+c_2u_{11}-\partial u_{21}/\partial t=0,\
(x,t)\in D_T^{(1)},\
a_{k2}\Delta u_{k2}-\partial u_{k2}/\partial t=0\quad (k=1,2),\qquad
(x,t)\in D_T^{(2)}
\end{gathered}
\tag{1}
]

with initial conditions for (t=0) for (u_{kl}(x,t)), linear boundary conditions for (u_{k2}) on (\Gamma^{(2)})—the outer boundary of the external medium (D_T^{(2)})—and conjugation conditions on the surface (\Gamma^{(1)}) of the cell membrane (D_T^{(1)}), immersed in the medium (D_T^{(2)}):

[
(-1)^l a_{kl}\partial u_{kl}(x,t)/\partial N_l(x,t)-h_k\bigl(u_{k2}(x,t)-u_{k1}(x,t)\bigr)=0,
\qquad
k,l=1,2,\quad (x,t)\in \Gamma^{(1)}.
\tag{2}
]

Here (u_{kl}(x,t)) in (1) is the concentration of the substance (S_k) inside ((l=1)) and outside ((l=2)) the cell. (S_1) is the substance entering the cell (D_T^{(1)}) from the external medium (D_T^{(2)}) and supplying the organism with energy. Under enzymatic transformations, part of (S_1) passes into low-molecular substances (S_2), which are removed from the cell, while from another part of (S_1) the cell synthesizes its specific substances (S_3). The constants (c_i>0) characterize the ability of the enzymes of the cell biomass to transform (S_1) into (S_i) for (i=2,3). (a_{kl}>0) ((k,l=1,2)) is the diffusion coefficient of the substance (S_k) inside ((l=1)) and outside ((l=2)) the cell. The constant (h_k>0) in (2) characterizes the permeability of the cell membrane (\Gamma^{(1)}) for the substance (S_k) ((k=1,2)). (N_l(x,t)) is the normal, internal with respect to (D_T^{(l)}), to the section (\Gamma^{(1)}\cap{t=t>0}).

We shall consider a more general problem for a system of parabolic equations with discontinuous coefficients of a special kind, preserving the specificity of the system (1)—(2):

[
\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}
-
\partial u_{kl}/\partial t
=
f_{kl}(x,t),
\quad
k=1,2,\ldots,m;\ l=1,2,\quad (x,t)\in D_T^{(l)}
\tag{3}
]

with supplementary conditions

[
u_{kl}(x,0)=f_{kl}^{(1)}(x),\qquad
x\in \Omega^{(l)}=\overline{D_T^{(l)}}\cap{t=0},\qquad
k=1,2,\ldots,m;\ l=1,2;
\tag{4}
]

[
a_k^{(k)}(x,t)\frac{\partial u_{k2}(x,t)}{\partial \nu_k(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(x,t)}
+
b_i^{(k)}(x,t)u_{i2}(x,t)
\right)
+
f_k^{(2)}(x,t),
\quad
(x,t)\in \Gamma^{(2)},\quad k=1,2,\ldots,m;
\tag{5}
]

[
(-1)^l d_{kl}^{(kl)}(x,t)\frac{\partial u_{kl}(x,t)}{\partial \nu_{kl}(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}
\left[
(-1)^{l+1}d_{il}^{(k)}(x,t)\frac{\partial u_{il}(x,t)}{\partial \nu_{il}(x,t)}
+
\sum_{j=1}^{2} h_{ij}^{(kl)}(x,t)u_{ij}(x,t)
\right]
+
f_{kl}^{(3)}(x,t),
\tag{6}
]

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

Throughout what follows we shall use the notation and definitions of works ((^{2-4})). (D_T^{(1)} \cup \Gamma^{(1)} \cup D_T^{(2)} = D_T) is a bounded domain of ((n+1))-dimensional Euclidean space ((x,t)=(x_1,x_2,\ldots,x_n,t)), lying between the two hyperplanes (t=0) and (t=T>0), having as its lower base the domain (\Omega^{(1)} \cup \Omega^{(2)}) and as its lateral boundary the closed surface (\Gamma^{(2)}); moreover (D_T^{(1)}) is an interior subdomain of (D_T) with lateral boundary surface (\Gamma^{(1)}), separating (D_T^{(1)}) and (D_T^{(2)}). The closed surfaces (\Gamma^{(l)}) ((l=1,2)), of Lyapunov type, the tangent planes to which are nowhere orthogonal to the axis (Ot), are situated between the hyperplanes (t=0) and (t=T) and do not intersect one another, being separated from each other by a distance not less than (d>0). On (\Gamma^{(1)}) there are prescribed fields of directions with unit vectors (\nu_{il}(x,t)) ((i=1,2,\ldots,m;\ l=1,2)), lying in the section (\Omega_t^{(l)}=D_T^{(l)}\cap{\tau=t}) and forming acute angles not exceeding (\pi/2-d_0) ((d_0>0)) with the normal (N_{1l}(x,t)), interior with respect to (\Omega_t^{(l)}), at the point ((x,t)\in\Gamma^{(1)}). On (\Gamma^{(2)}) there are prescribed fields of directions with unit vectors (\nu_i(x,t)) ((i=1,2,\ldots,m)), lying in the section (\Omega_t^{(2)}) and forming acute angles not exceeding (\pi/2-d_0) with the interior normal (N_2(x,t)) at the point ((x,t)\in\Gamma^{(2)}).

In conditions (5), (6) the inequalities are always assumed to hold

[
a_k^{(k)}(x,t)\ge a_0>0,\quad b_k^{(k)}(\bar x,t)\ge 0,\quad d_{kl}^{(k)}(\bar x,t)\ge \delta>0,\quad h_{kl}^{(kl)}(\bar x,t)\ge 0,
]

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

§ 1. Suppose that for (3)—(6) the additional conditions are satisfied

[
b_{ij}^{(kl)}(x,t)\equiv c_i^{(kl)}(x,t)\equiv 0 \text{ in (3)},\quad
a_i^{(k)}(x,t)\equiv b_i^{(k)}(x,t)\equiv 0 \text{ in (5)},
]

[
d_{il}^{(k)}(x,t)\equiv h_{ij}^{(kl)}(x,t)\equiv 0 \text{ in (6)}\quad
\text{for } i=k+1,\ldots,m.
\tag{7}
]

With the aid of the maximum principle and Vyborny’s theorem ((^5)) one proves the uniqueness of the solution of the system (3)—(7).

Theorem 1. Suppose that for problem (3)—(7) the following conditions are satisfied: 1) all coefficients and functions entering (3)—(6) are continuous in their domains of definition, and
[
c_k^{(kl)}(x,t)\le c<+\infty,\quad (x,t)\in\overline D_T^{(l)},\quad k=1,2,\ldots,m;\quad l=1,2;
]
2) equations (3) are uniformly parabolic in (\overline D_T^{(l)}) with parabolicity constant (M_0>0).

Then the solution of problem (3)—(7) is unique in the class of functions (u_{kl}(x,t)), (k=1,2,\ldots,m;\ l=1,2), satisfying (3)—(7) and continuous on (\overline D_T^{(l)}).

If one somewhat narrows the class of systems (3)—(7) by the conditions

[
b_{ij}^{(kl)}(x,t)\equiv 0 \text{ in (3)};\quad
a_i^{(k)}(x,t)\equiv 0 \text{ in (5)};\quad
d_{il}^{(k)}(x,t)\equiv 0 \text{ in (6)}
\tag{8}
]

[
\text{for } i=1,2,\ldots,k-1,
]

then we obtain a class of systems possessing a number of interesting properties, following from the maximum principle, and allowing one, by means of the methods of the author and V. N. Maslennikova ((^6)), to give, in particular, an a priori estimate for the maximum modulus of the solution of problem (3)—(8).

Theorem 2. Suppose that for the solution of problem (3)—(8) conditions 1), 2) of Theorem 1 are satisfied and, in addition: 3) the coefficients of equations (3) belong to the class (H^{0,\alpha,\alpha/2}(\overline D_T^{(l)})) (see ((^3))), their maxima of moduli being bounded by a constant (A), and, moreover,
[
c_k^{(kl)}(x,t)\le c<0,\quad (x,t)\in\overline D_T^{(l)};
]
4) the functions entering the right-hand sides of conditions (5), (6) are bounded in modulus by a constant (B); 5) the functions entering the left-hand sides of (5), (6) belong to the class (H_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}); 6) the surfaces (\Gamma^{(l)}) ((l=1,2)) are of class (\Pi_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}), (0<\alpha\le 1).

Then for the solution (u_{kl}(x,t)) of problem (3)—(8), continuous on (\overline D_T^{(l)}), the estimate
[
|u_{kl}|_{0}^{D_T^{(l)}}\le C(d,d_0,\delta,a_0,A,B,c),
]
holds, where (C(\ldots)>0) is a constant.

§ 2. Existence of a solution of (3)—(7).

Theorem 3. Suppose that for problem (3)—(7) conditions 1), 2) of Theorem 1 are satisfied. Suppose, in addition: 7) the surfaces (\Gamma^{(l)}) ((l=1,2)) are of class (L_{1,1,(1+\beta)/2}^{1,\beta,\beta/2}) ((0<\alpha<\beta\leqslant 1)); 8) the coefficients in (3) have an ((\alpha))-norm (see ((2^4))) bounded by the constant (M_1); 9) the initial functions from (4) have bounded ((2+\alpha))-norms, and the coefficients entering into (5), (6) have ((1+\alpha))-norms bounded by the constant (M_2); 10) the initial conditions (4) and boundary conditions (5), (6) are compatible by virtue of (3) on the edges (\Gamma^{(s)}\cap\Omega^{(l)}) ((l=1,2) for (s=1) and (l=2) for (s=2)).

Then there exists a solution (u_{kl}(x,t)) ((k=1,2,\ldots,m;\ l=1,2)) of problem (3)—(7) belonging to the class
[
H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\bigl(\overline{D_T^{(l)}}\bigr).
]

The proof of the existence of a solution of problem (3)—(7) is carried out by the classical method of continuation with respect to a parameter (see ((^7))), if one uses the ((2+\alpha))-a priori estimate for the solution of problem (3)—(6), established by Theorem 4, and the existence of a solution from the class (H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}(\overline{D_T^{(l)}})) of the system ((3^0)), (4)—(7), established by Theorem 5 (equation ((3^0)) is obtained from (3) by putting in (3) (b_{ij}^{(kl)}(x,t)\equiv c_i^{(kl)}(x,t)\equiv 0), (a_{ij}^{(kl)}(x,t)\equiv a_{ij}^{(kl)}), where the matrix of constant coefficients (|a_{ij}^{(kl)}|) is symmetric and positive definite).

Theorem 4. Suppose that all the conditions of Theorem 3 are satisfied (with 7) replaced by 6) from Theorem 2), and moreover
[
\det |a_j^{(k)}(x,t)|\ne 0,\quad (x,t)\in\Gamma^{(2)},\qquad
\det |d_{jl}^{(k)}(x,t)|\ne 0,\quad (x,t)\in\Gamma^{(1)},\quad l=1,2.
]

Suppose problem (3)—(6) has a solution (u_{kl}(x,t)) for which
[
\left|u_{kl}\right|_{2+\alpha}^{D_T^{(l)}}<+\infty.
]

Then the estimate holds
[
\begin{aligned}
\left|u_{kl}\right|{2+\alpha}^{D_T^{(l)}} \leqslant\;&
C\bigl(D_T^{(l)},d,d_0,\delta,a_0,M_0,M_1,M_2\bigr)
\max\Bigl(
\left|f_{ij}\right|{\alpha}^{D \}^{(j)}
&\quad +\left|f_{ij}^{(1)}\right|{2+\alpha}^{\Omega^{(j)}}
+\left|f_j^{(2)}\right|}^{\Gamma^{(2)}
+\left|f_{ij}^{(3)}\right|{1+\alpha}^{\Gamma^{(1)}}
+\left|u\Bigr).}\right|_{0}^{D_T^{(j)}
\end{aligned}
\tag{9}
]

Remark. If uniqueness holds for problem (3)—(6) (see Theorems 1, 2), then, by virtue of the boundedness of (D_T), in estimate (9) one may omit the term (\left|u_{ij}\right|_{0}^{D_T^{(j)}}) appearing on the right.

Theorem 4 is proved by the same method by which, in the works of the author and V. N. Maslennikova ((^8)) (see also ((^{2,4}))), a ((2+\alpha))-a priori estimate was established for the solution of a problem with an oblique derivative for a parabolic equation of second order in a noncylindrical domain.

Theorem 5. Suppose that for problem ((3^0)), (4)—(6), (8) all the conditions of Theorem 3 are satisfied. Then there exists a solution (u_{kl}(x,t)) ((k=1,2,\ldots,m;\ l=1,2)) of problem ((3^0)), (4)—(7) belonging to the class
[
H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}\bigl(\overline{D_T^{(l)}}\bigr).
]

The proof of Theorem 5 is carried out by the method of potentials with the aid of a number of results (see Lemmas 1—6) of the theory of special Pani heat potentials ((^9)), quite analogous to the theory of ordinary heat potentials from ((^{3,10,11})). If on (\Gamma\equiv\Gamma^{(2)}), the lateral boundary of the domain (D_T), a field of directions (\nu(x,t)) satisfying (7) is prescribed, and (P(\bar{x},t;y,\tau)) is the special fundamental solution of Pani ((^9,\S 4)), corresponding to (\nu(x,t)) and to the parabolic operator with constant coefficients
[
\sum_{i,j=1}^{n} a_{ij}\frac{\partial^2}{\partial x_i\partial x_j}-\frac{\partial}{\partial t},
]
then we consider the special heat surface potentials

potentials for ((\bar x,t)\in D_T,\ (x,t)\in\Gamma)

[
P[\varphi]\equiv P(\bar x,t)=\int_0^t d\tau \iint_{\Gamma_\tau} P(\bar x,t;y,\tau)\varphi(y,\tau)\,d\sigma_y(\tau),
\tag{10}
]

[
Q[\varphi]\equiv Q(\bar x,t)=\frac{\partial P(\bar x,t)}{\partial \nu(x,t)}
]

and their direct values on (\Gamma): (\bar P[\varphi]=\bar P(x,t)), (\bar Q[\varphi]=\bar Q(x,t)), obtained from (10) for ((\bar x,t)\equiv(x,t)\in\Gamma).

Lemma 1. If (\Gamma) is of type (Л_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}) ((0<\alpha\leq 1)), ([\varphi]_0^\Gamma<+\infty), ([\nu]_0^\Gamma<+\infty), then (P[\varphi]\in H^{0,\alpha_0,1/2}(\bar D_T)) ((\alpha_0) is any number for which (0<\alpha_0<1)), and the Hölder constants have the form ((C)[\varphi]_0^\Gamma). Moreover, (P(\bar x,0)\equiv 0).

Lemma 2. If (\Gamma) is of type (Л_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}), (|\nu|_\beta^\Gamma<+\infty) ((0<\alpha\leq\beta\leq 1)) and ([\varphi]_0^\Gamma<+\infty), then (\bar Q[\varphi]\in H^{0,\alpha^,\alpha^/2}(\Gamma)) ((\alpha^*=\min(\alpha,\beta')), where (\beta') is any number for which (0<\beta'<\beta)), and the Hölder constants have the form ((C)[\varphi]_0^\Gamma), and (\bar Q(x,0)\equiv 0).

Lemma 3. Let (\Gamma) be of type (Л_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}), (|\nu|_\beta^\Gamma<+\infty) ((0<\alpha\leq\beta\leq 1)), and let ((\bar x,t)\in\Omega_t) tend to (x), ((x,t)\in\Gamma), along the inner normal to (\Gamma_t). Then for continuous (\varphi)

[
\lim_{(\bar x,t)\to(x,t)\in\Gamma_t} Q(\bar x,t)
=\bar Q(x,t)-2^{\,n-1}\pi^{n/2}(\det |a^{ij}|)^{-1/2}\varphi(x,t),
]

where (|a^{ij}|) is the matrix inverse to (|a_{ij}|). If (|\varphi|_\alpha^\Gamma<+\infty) and (\varphi(x,0)\equiv 0),

[
\left|Q(\bar x,t)-\bar Q(x,t)+2^{\,n-1}\pi^{n/2}(\det |a^{ij}|)^{-1/2}\varphi(x,t)\right|
\leq (C)|\varphi|_\alpha^\Gamma|\bar x-x|^{\alpha^0}
]

[
(\alpha^0=\alpha \text{ for } 0<\alpha<1;\ \alpha^0=\alpha_0 \text{ for } \alpha=1).
]

Lemma 4. Let (\Gamma) be of type (Л_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}), (|\nu|{1+\beta}^\Gamma<+\infty), (|\varphi|\alpha^\Gamma<+\infty) ((0<\alpha\leq\beta\leq 1)), and (\varphi(x,0)\equiv 0). Then (Q[\varphi]\in H_{1,\alpha',\alpha^/2}^{0,1,(1+\alpha^)/2}(\Gamma)), and the Hölder constants have the form ((C)|\varphi|_\alpha^\Gamma), and (\bar Q(x,0)\equiv 0).

Lemma 5. Let (\Gamma) be of type (Л_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}), (|\nu|\beta^\Gamma<+\infty), (|\varphi|\alpha^\Gamma<+\infty) ((0<\alpha\leq\beta\leq 1)), and (\varphi(x,0)\equiv 0). Then (P[\varphi]\in H_{1,\alpha',\alpha'/2}^{0,1,(1+\alpha')/2}(\bar D_T)), and the Hölder constants have the form ((C)|\varphi|_\alpha^\Gamma), and (P(x,0)\equiv 0).

Lemma 6. Let (\Gamma) be of type (Л_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}), (|\nu|{1+\beta}^\Gamma<+\infty), (|\varphi|^\Gamma), and (P(x,0)\equiv 0).}^\Gamma<+\infty), (\varphi(x,0)\equiv 0) ((0<\alpha\leq\beta\leq 1)). Then (P[\varphi]\in H_{1,1,(1+\alpha')/2}^{1,\alpha',\alpha'/2}(\bar D_T)), and the Hölder constants have the form ((C)|\varphi|_{1+\alpha

Remark. If (\Gamma) is of type (Л_{1,1,(1+\beta)/2}^{1,\beta,\beta/2}), where (0<\alpha<\beta\leq 1), then, under the remaining assumptions of Lemma 6, (P[\varphi]\in H_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}(\bar D_T)).

Moscow State University
named after M. V. Lomonosov

Received
29 X 1965

CITED LITERATURE

1. V. I. Skobelkin, A. A. Boldin, DAN, 145, No. 6, 1396 (1962).
2. L. I. Kamynin, V. N. Maslennikova, DAN, 153, No. 3, 526 (1963).
3. L. I. Kamynin, DAN, 160, No. 2, 271 (1965).
4. L. I. Kamynin, V. N. Maslennikova, DAN, 160, No. 3, 527 (1965).
5. R. Vyborny, DAN, 117, No. 4, 563 (1957).
6. L. I. Kamynin, V. N. Maslennikova, Siberian Math. Journal, 2, No. 3, 384 (1961).
7. S. N. Bernstein, Math. Ann., 69, 82 (1910).
8. L. I. Kamynin, V. N. Maslennikova, Siberian Math. Journal, 7, No. 1 (1966).
9. M. Ragni, Ann. Scuola norm. sup. Pisa, ser. III, II, fasc. I—II, 73 (1957).
10. L. I. Kamynin, Differential Equations, 1, No. 6, 799 (1965).
11. L. I. Kamynin, Differential Equations, 2, No. 5 (1966).