MATHEMATICS

Ya. I. ZHITOMIRSKII

ON THE CAUCHY PROBLEM FOR A PARABOLIC EQUATION OF SECOND ORDER WITH VARIABLE COEFFICIENTS

(Presented by Academician I. G. Petrovskii, 15 V 1957)

Consider the equation

[
\frac{\partial u}{\partial t}+Lu=f(x,t),
\tag{1}
]

where

[
L=-\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}a_{ij}(x)\frac{\partial}{\partial x_j}
+\sum_{i=1}^{n}b_i(x)\frac{\partial}{\partial x_i}+c(x),\quad
x=(x_1,\ldots,x_n).
]

Introduce the notation:

[
\Delta(x)=\det|a_{ij}(x)|,\quad
\Delta_i(x)=
\left|
\begin{array}{cccccc}
a_{11}\ldots a_{1,i-1} & b_1 & a_{1,i+1}\ldots a_{1n}\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\
\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\
a_{n1}\ldots a_{n,i-1} & b_n & a_{n,i+1}\ldots a_{nn}
\end{array}
\right|.
]

Suppose that the coefficients of the operator (L) satisfy the following conditions:

[
a_{ij}(x)=a_{ji}(x)
\tag{2}
]

for all (x),

[
\sum_{i,j=1}^{n} a_{ij}(x)\xi_i\xi_j \geq \alpha^2\sum_{i=1}^{n}\xi_i^2
\quad(\alpha>0)
\tag{3}
]

and the coefficient (c(x)) is bounded from below:

[
c(x)>-c \quad (c>0).
\tag{4}
]

Let there exist a differentiable function (g(x)) such that

[
\frac{\partial g}{\partial x_i}=\frac{\Delta_i(x)}{\Delta(x)}.
\tag{5}
]

To equation (1) we append the initial condition

[
u(x,0)=0.
\tag{1'}
]

In the present paper we study questions of existence and uniqueness of the solution of the Cauchy problem (1)—(1′) in the whole space ((x_1,\ldots,x_n)) in the class of rapidly increasing functions.

Following M. I. Vishik ((^{1})), we define a generalized solution of the Cauchy problem (1)—(1′). For this we shall need to introduce into consideration certain Hilbert spaces of functions.

Let us first consider the space (\Omega_L) of finite twice differentiable functions with the scalar product

[
(u,\ v)=\int_{-\infty}^{\infty}\cdots\int u(x,\ t)v(x,\ t)e^{-g(x)}dx.
\tag{6}
]

We shall show that the operator (L) is symmetric on (\Omega_L). Let (\Omega) be a domain in the space ((x_1,\ldots,x_n)); let (\Gamma) be its boundary; and let (\mathbf n) be the outward normal to (\Gamma). Putting, in the well-known Ostrogradsky formula,

[
\int_{\Omega}\cdots\int \operatorname{div}\mathbf R\,dx
=
\int_{\Gamma}\cdots\int(\mathbf R,\mathbf n)\,d\sigma
]

[
((\mathbf R,\mathbf n)\text{ is the ordinary scalar product of two vectors})
]

[
\mathbf R=\left{v(x)e^{-g(x)}\sum_{j=1}^{n}a_{ij}(x)\frac{\partial u}{\partial x_j}\right},
]

and taking for (\Omega) a sphere of radius (r), then letting (r) tend to infinity, we easily obtain the formula

[
-\int_{-\infty}^{\infty}\cdots\int
\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}a_{ij}(x)\frac{\partial u}{\partial x_j}v(x)e^{-g(x)}dx
=
\tag{7}
]

[

\int_{-\infty}^{\infty}\cdots\int
\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial u}{\partial x_j}\frac{\partial v}{\partial x_i}e^{-g(x)}dx
-
\int_{-\infty}^{\infty}\cdots\int
\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial u}{\partial x_j}v(x)e^{-g(x)}
\frac{\Delta_i(x)}{\Delta(x)}dx.
]

From (7) it follows that

[
(Lu,\ v)=
\int_{-\infty}^{\infty}\cdots\int
\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial u}{\partial x_j}\frac{\partial v}{\partial x_i}e^{-g(x)}dx+
]

[
+\sum_{j=1}^{n}\int_{-\infty}^{\infty}\cdots\int
\left(
-\sum_{i=1}^{n}a_{ij}(x)\frac{\Delta_i(x)}{\Delta(x)}
+b_j(x)
\right)
\frac{\partial u}{\partial x_j}v(x)e^{-g(x)}dx+
]

[
+\int_{-\infty}^{\infty}\cdots\int c(x)u(x)v(x)e^{-g(x)}dx.
]

But, by virtue of the definition of the quantities (\Delta_i(x)) and (\Delta(x)), it is obvious that

[
\sum_{i=1}^{n}a_{ij}(x)\frac{\Delta_i(x)}{\Delta(x)}=b_j(x)\quad (j=1,\ldots,n).
]

Therefore

[
(Lu,\ v)=
\int_{-\infty}^{\infty}\cdots\int
\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial u}{\partial x_j}\frac{\partial v}{\partial x_i}e^{-g(x)}dx
+
\int_{-\infty}^{\infty}\cdots\int c(x)u(x)v(x)e^{-g(x)}dx.
\tag{8}
]

Interchanging in (8) the roles of the functions (u(x)) and (v(x)), and taking (2) into account, we obtain the symmetry of the operator (L) on (\Omega_L):

[
(Lu,\ v)=(u,\ Lv).
\tag{9}
]

From formula (8) and conditions (3), (4) it follows that ((Lu,u)>-c(u,u)). Making in (1) the substitution of the unknown function (u(x,t)=e^{k_1t}v(x,t)) and choosing (k_1) so that (k=k_1-c>0), we obtain that the operator (L_1=L+k) satisfies the condition

[
(L_1u,\ u)>k(u,\ u).
\tag{10}
]

Therefore one may assume that (L) also satisfies condition (10).

We now introduce in the space (\Omega_L) a new metric by the formula

[
{u,u}'=(Lu,u).
\tag{11}
]

Complete the space (\Omega_L) with respect to the metrics (6) and (11). The resulting complete Hilbert spaces will be denoted respectively by (H) and (H').

Since the bilinear form ((Lu,v)) is bounded in the metric (11), it can be realized as the bilinear form of some bounded symmetric operator (\widetilde L) in (H')

[
(Lu,v)={\widetilde Lu,v}'.
\tag{12}
]

We now consider the operator (S=\dfrac{\partial}{\partial t}+L), defined on the space of functions (\Omega_S) satisfying the following conditions:

1) (u(x,0)=0);

2) (u(x,t)\in\Omega_L) for all (t\in[0,T]);

3) (\dfrac{\partial u}{\partial t}) exists in (H) for (t\in[0,T]), and moreover
[
\int_0^T \left(\frac{\partial u}{\partial t},\frac{\partial u}{\partial t}\right)\,dt<\infty;
]

4)
[
\int_0^T (Lu,Lu)\,dt<\infty.
]

In the space of functions (\Omega_S) we introduce two metrics by the formulas

[
[u,v]=\int_0^T (u,v)\,dt;
\tag{13}
]

[
{u,v}_1=\int_0^T {u,v}'\,dt.
\tag{14}
]

Complete the space (\Omega_S) with respect to the metrics (13) and (14). The resulting complete Hilbert spaces will be denoted respectively by (\mathfrak A) and (\mathfrak A'). Since

[
\left|\left[\frac{\partial u}{\partial t},v\right]\right|
\le
\left[\frac{\partial u}{\partial t},\frac{\partial u}{\partial t}\right]\cdot [v,v]
\le
C_1\int_0^T (Lv,v)\,dt
\le
C_2{v,v}_1,
]

the bilinear form (\left[\dfrac{\partial u}{\partial t},v\right]) can be realized in (\mathfrak A') in the form of the form of some operator (\widetilde T)

[
\left[\frac{\partial u}{\partial t},v\right]={\widetilde Tu,v}_1.
]

At the same time

[
[Lu,v]=\int_0^T (Lu,v)\,dt
=\int_0^T {\widetilde Lu,v}'\,dt
={\widetilde Lu,v}_1.
]

Thus,

[
[Su,v]={\widetilde Tu,v}_1+{\widetilde Lu,v}_1.
\tag{15}
]

Now assuming that (v(x,t)) belongs to the space (\Omega^), consisting of functions (v(x,t)) belonging to (\Omega_S) and satisfying the condition (v(x,T)=0) (obviously, (\Omega^) is dense in (\Omega_S) in the metric (14)), one may in (15) transfer the operator (\widetilde T) to the function (v(x,t))

[
[Su,v]={u,\widetilde T^v}_1+{u,\widetilde L^v}_1.
\tag{16}
]

From (16) it follows that, if (u(x,t)) is an ordinary solution of equation (1), then

[
[f,v]={u,(\widetilde T^+\widetilde L^)v}_1.
\tag{17}
]

By a generalized solution of problem (1)—(1′) we shall mean a function (u(x,t)) belonging to the space (\mathfrak{A}'), for which equality (17) holds for all (v(x,t)) from the space (\Omega^*) ((^1)).

Since the operator (L) satisfies conditions (9) and (10) (symmetry and positive definiteness), it follows, by the results of ((^1)) (see also ((^2))), that the following theorem holds.

Theorem. For any function (f(x,t)) satisfying the condition
[
\int_0^T (f,f)\,dt<\infty,
]
there exists a generalized solution of problem (1)—(1′), and it is unique.

Let us consider, as an example, an equation of the form (1), in which
[
Lu=-\Delta u+\sum_{i=1}^n b_i(x_i)\frac{\partial u}{\partial x_i}+c(x)u,
]
where (\Delta) is the (n)-dimensional Laplace operator; (b_i) depends only on (x_i); (x=(x_1,\ldots,x_n)). It is easy to see that conditions (2)—(5) are satisfied here.

In this case
[
g(x)=\sum_{i=1}^n \int_0^{x_i} b_i(y)\,dy.
]
Then for any function (f(x,t)) satisfying the condition
[
\int_0^T \int_{-\infty}^{\infty}\cdots\int f^2(x,t)e^{-g(x)}\,dx\,dt<\infty,
]
there exists a generalized solution of the Cauchy problem (1)—(1′), and it is unique.

In particular, for (n=1), (b(x)=x^{2m-1}), we obtain uniqueness of the solution of the Cauchy problem (1)—(1′) in the class of functions (u(x,t)) satisfying the conditions
[
|u(x,t)|\leq A_1 e^{A_2|x|^{2m-\varepsilon}},\qquad
\left|\frac{\partial u(x,t)}{\partial x}\right|\leq A_3 e^{A_4|x|^{2m-\varepsilon}},
]
for any (\varepsilon>0); (0\leq t\leq T); (A_1,A_2,A_3,A_4) are constants.

These examples show that the uniqueness class for the solution of the Cauchy problem (1)—(1′) can be extended without bound (in the sense of the admissible growth as (|x|\to\infty) of the functions entering it), depending on the behavior as (|x|\to\infty) of the coefficients of equation (1).

The mixed problem for equation (1) for the half-space (x_i>0) is posed as follows: one seeks a solution of equation (1) satisfying condition (1′) and the boundary condition
[
u(x_1,\ldots,x_n,t)\big|_{x_i=0}=0. \tag{1″}
]

Problem (1)—(1′)—(1″) is reduced in the usual way ((^1)) to problem (1)—(1′). For this it is necessary to impose on the functions of the space (\Omega_L) the additional restriction (u(x,t)=0) for (x_i\leq 0). Then, as above:

Theorem. For any function (f(x,t)) satisfying the condition
[
\int_0^T (f,f)\,dt<\infty,
]
there exists a generalized solution of problem (1)—(1′)—(1″), and it is unique.

The examples analyzed above show that in this case as well the uniqueness class for the solution of problem (1)—(1′)—(1″) contains functions whose admissible growth as (|x|\to\infty) can increase without bound as the growth as (|x|\to\infty) of the coefficients of equation (1) increases.

Moscow State University
named after M. V. Lomonosov

Received
11 V 1957

REFERENCES

1. M. I. Vishik, Matem. sbornik, 39 (81), 1 (1956).
2. M. I. Vishik, O. A. Ladyzhenskaya, Uspekhi matem. nauk, 11, no. 6 (72) (1956).