MATHEMATICS

Yu. N. CHEREMNYKH

ON A THEOREM OF THE QUALITATIVE THEORY OF PARABOLIC EQUATIONS

(Presented by Academician I. G. Petrovskii, July 20, 1959)

Consider the equation

[
\frac{\partial^2 u}{\partial x^2}
=
a(t,x)\frac{\partial u}{\partial t}
+
b(t,x)\frac{\partial u}{\partial x}
+
c(t,x)u,
\tag{1}
]

given in the closed rectangular domain (\overline G=(0\leq t\leq T\leq 1;\ \delta\leq x\leq 1-\delta,\ 1/2>\delta>0)) of the plane ((t,x)).

The coefficients of equation (1) satisfy in (\overline G) the following conditions:
1) all coefficients are bounded in modulus by one;
2) (a(t,x)\geq a_0>0,\ c(t,x)\geq 0);
3) (a(t,x), b(t,x), c(t,x)\in C^{(2)}) with respect to (x) and are bounded in modulus together with all required derivatives;
4) the coefficient (a(t,x)) has a derivative with respect to (t);
5) (|\partial a(t,x)/\partial t|\leq 1,\ |\partial b(t,x)/\partial x|\leq 1).

The solution (u(t,x)) of equation (1) will be assumed to be twice continuously differentiable in (\overline G), to belong to (C^{(4)}) with respect to (x), and such that (|u(t,x)|<1).

Let some positive number (\xi<\dfrac{1-2\delta}{4}) be given.

Introduce the following notation:
(\Sigma=(\delta\leq x\leq 1-\delta,\ t=T));
(\xi_1=(\delta\leq x<\delta+\xi,\ t=T)),
(\xi_2=(\delta+\xi\leq x<\delta+2\xi,\ t=T)),
(\xi_3=(1-\delta-2\xi<x\leq 1-\delta-\xi,\ t=T)),
(\xi_4=(1-\delta-\xi<x\leq 1-\delta,\ t=T));
(\Sigma^=\Sigma\setminus \xi_1\setminus \xi_2\setminus \xi_3\setminus \xi_4);
(\Pi_T=(0<t<T)), (\Pi_2=(T/2<t<T)),
(\Pi^{*}=(t_1<t<T)) ((T/2\leq t_1<T));
(\gamma=T-t_1);
(\Delta_1=(\delta<x<\delta+\xi,\ T/2<t<T)),
(\Delta_2=(1-\delta-\xi<x<1-\delta,\ T/2<t<T)),
((G\cap \Pi_2)\setminus \Delta_1\setminus \Delta_2=G_{2\Delta});
((1-2\delta)T=\sigma).

Put, as was done in ((^1)),

[
G_+={(t,x)\in G,\ u(t,x)>0},\qquad
G_-={(t,x)\in G,\ u(t,x)<0}.
]

We shall call essential those components of the sets (G_+) and (G_-) which have limit points on both sides of the strip (\Pi_T).

Suppose that the solution (u(t,x)) has in the domain (G) (N) essential components (g_i) ((i=1,\ldots,N)). Suppose that (N_1) essential components have limit points on (\Sigma^*).

The proof of the following lemma is, in idea, analogous to the proof of Theorem 2.6.1 of ((^1)).

Lemma. There exist absolute constants (M_1) and (M_2) such that, if

[
N_1 \geq
\frac{4M_1^{1/2}M_2^{1/2}\sigma}{\gamma^{3/2}t_1^{3/2}},
]

then for every level line (l) connecting both sides of the strip (\Pi^{**}), the inequality

[
|u|_l
<
2^{-\frac{t_1^3 N_1^2}{4M_1\sigma^2}}
]

holds.

((M_1=80^2,\ M_2>640^2)—absolute constants, respectively, of Theorem 2.3.1 and Lemma 2.5.1 of [1]).

In what follows we shall assume that

[
N_1 \geq \frac{32 M_1^{1/2} M_2^{1/2}(1-2\delta)}{T^2}.
\tag{2}
]

(1^\circ.) In the case when there are essential components having limit points on (\xi_2) (on (\xi_1) and (\xi_2)) and on (\xi_3) (on (\xi_3) and (\xi_4)), the dependence between (\max_{\Sigma^*}|u(T,x)|) and the number (N_1) is established by a direct application of Theorem 2.6.1 of [1].

(2^\circ.) Consider the case when among the essential components there are some that have limit points on (\xi_2) (on (\xi_1) and (\xi_2)), but there are no essential components having limit points on (\xi_3) and (\xi_4), or, conversely, there are essential components having limit points on (\xi_3) (on (\xi_3) and (\xi_4)), and there are no essential components having limit points on (\xi_1) and (\xi_2), as well as the case when (N_1=N).

Let (\max_{\Sigma^}|u(T,x)|=2\varepsilon_0). Denote
(S_1={(t,x)\in \Sigma^, |u(t,x)|=2\varepsilon_0}),
(S_2={(t,x)\in \Sigma^*, |u|=\varepsilon_0}). Obviously, the sets (S_1) and (S_2) are nonempty.

The following subcases are possible:

(2^\circ_1.) The set (S_2) contains at least one point ((T,x')) such that the level line (l') passing through it connects both sides of the strip (\Pi_2) (the set (S_1) may have no such point).

Put (t_1=T/2) (then, consequently, (\gamma=T/2)). Using the lemma, we find that

[
\max_{\Sigma^*}|u(T,x)|=2\varepsilon_0
<
2\cdot 2^{-\frac{1}{32}\frac{T N_1^2}{(1-2\delta)^2 M_1}} .
\tag{3}
]

(2^\circ_2.) The level lines passing through points of the sets (S_1) and (S_2) have no common points with the straight line (t=T/2).

Let the point ((T,x_2)\in S_1). Without loss of generality, we may assume that (u(T,x_2)) is positive. Let ((T,x_3)\in S_2) belong to the same component (G_+) to which ((T,x_2)) belongs. Denote by (l_{2\varepsilon_0}) and (l_{\varepsilon_0}) the level lines passing, respectively, through the points ((T,x_2)) and ((T,x_3)). The lines (l_{2\varepsilon_0}) and (l_{\varepsilon_0}) exit either through the left or through the right boundary of the domain (G). The two cases are symmetric; therefore in what follows we shall assume that the lines (l_{2\varepsilon_0}) and (l_{\varepsilon_0}) go to the left and that ((T,x_2)) is situated to the left of ((T,x_3)).

Obviously,

[
1-2\delta-3\xi > r_0 \geq \xi, \qquad \text{where } r_0=x_2-\delta-\xi .
]

On the basis of (2), for the domain (\overline{G}) one can indicate a constant (\widetilde{M}_2\geq 2), depending only on the coefficients (a(t,x)), (b(t,x)), (c(t,x)) and their derivatives, such that

[
\left|\frac{\partial^2 u}{\partial x^2}\right|
<
\frac{\widetilde{M}2(1-2\delta)^2 \max}}|u|
{t\left[\left(\frac{1-2\delta}{2}\right)^2-\left(\frac{1-2x}{2}\right)^2\right]^2}.
]

For the domain (\overline{G}_{2\Delta}) we have

[
\left|\frac{\partial^2 u}{\partial x^2}\right|
<
\frac{2\widetilde{M}2(1-2\delta)^2 \max}}|u|
{T \xi^2(1-2\delta-\xi)^2}.
\tag{4}
]

The set of values of the function (u(T,x)) on (\Sigma^*) is bounded by constants (\underline{m}) and (\overline{m}). By the Kronrod–Landis theorem [3], for almost all (\tau) of the interval ([\underline{m},\overline{m}]) the corresponding level sets
(L_\tau={(t,x)\in G,\ u(t,x)=\tau})

do not have points at which (\operatorname{grad} u=0). Denote by (\mathfrak{M}) the set of such (\tau^\in[\underline m,\bar m]) for which at no point ((t^,x^)\in L_{\tau^}) does (\operatorname{grad} u(t^,x^)) vanish.

Obviously, (\mu_1\mathfrak{M}=\bar m-\underline m). Suppose first that (2\varepsilon_0) and (\varepsilon_0) belong to (\mathfrak{M}). Then the lines (l_{2\varepsilon_0}) and (l_{\varepsilon_0}) are smooth curves. Denote by (\mathcal{T}) the set of points lying between (l_{2\varepsilon_0}) and (l_{\varepsilon_0}).

Put also (U=(x_1=\delta+\xi<x<x_2=\delta+\xi+r_0)), (\Omega=\mathcal{T}\cap U). Let the straight line (t=t_2) pass through the point belonging to (l_{\varepsilon_0}) and having the smallest (t)-coordinate in (\overline{G}_{2\Delta}).

We have

[
0=\iint_{\Omega} a(t,x)\frac{\partial u}{\partial t}\,dt\,dx
+\iint_{\Omega} b(t,x)\frac{\partial u}{\partial x}\,dt\,dx
+\iint_{\Omega} c(t,x)u\,dt\,dx
-\iint_{\Omega}\frac{\partial^2 u}{\partial x^2}\,dt\,dx
=I_1+I_2+I_3+I_4 .
]

Since

[
\begin{aligned}
I_1&>\varepsilon_0 a_0 r_0-3\varepsilon_0 r_0(T-t_2)-r_0(T-t_2),\
I_2&>(-3\varepsilon_0 r_0-6\varepsilon_0-2-r_0)(T-t_2),\
I_3&>-r_0(T-t_2),\qquad
I_4>-\frac{2\widetilde M_2(1-2\delta)^2}{T\xi^2(1-2\delta-\xi)^2}(T-t_2)r_0\,^*,
\end{aligned}
]

we obtain that

[
T-t_2>
\frac{\varepsilon_0 a_0 r_0}
{6\varepsilon_0 r_0+2+3r_0+6\varepsilon_0+
\dfrac{2\widetilde M_2(1-2\delta)^2}{T\xi^2(1-2\delta-\xi)^2}r_0}.
]

Obviously,

[
\frac{\varepsilon_0 a_0 r_0}
{6\varepsilon_0 r_0+2+3r_0+6\varepsilon_0+
\dfrac{2\widetilde M_2(1-2\delta)^2}{T\xi^2(1-2\delta-\xi)^2}}

\frac{\varepsilon_0 r_0 a_0}
{\dfrac{3\widetilde M_2(1-2\delta)^2}{T\xi^2(1-2\delta-\xi)^2}}.
]

Put

[
\gamma_0=
\frac{\varepsilon_0 r_0 a_0}
{\dfrac{3\widetilde M_2(1-2\delta)^2}{T\xi^2(1-2\delta-\xi)^2}}
]

(obviously, (\gamma_0<T/96)) and (t_1^0=T-\gamma_0); then

[
T-t_2>\gamma_0 .
]

We shall show that

[
N_1<
\frac{4M_1^{1/2}M_2^{1/2}(1-2\delta)T}{\gamma_0^{3/2}t_1^{0\,3/2}} .
\tag{5}
]

Suppose the contrary. Then, by the lemma,

[
\frac{3\widetilde M_2(1-2\delta)^2\gamma_0}
{T\xi^2(1-2\delta-\xi)^2a_0r_0}
=\varepsilon_0
<
2^{-\frac{t_1^{0\,3}N_1^2}{4M_1\xi^2}},
]

whence it follows that

[
\log_2
\frac{T\xi^2(1-2\delta-\xi)^2a_0r_0}

\frac{4M_2}{\gamma_0^3},
]

which is impossible. Inequality (5) is proved.

(^*) The integrals (I_1) and (I_2) are estimated with the aid of Green’s formula. In estimating the integral (I_4), estimate (4) is used.

It is obvious that

[
N_1^{2/3}<
\frac{3\cdot 2^{4/3}M_1^{1/3}\widetilde M_2(1-2\delta)^{3/8}T^{1/3}M_2^{1/3}}
{t_1^0\varepsilon_0a_0\xi^2T(1-2\delta-\xi)^2r_0}
<
\frac{2^{16/3}M_1^{1/3}M_2^{1/3}\widetilde M_2(1-2\delta)^{2/3}}
{\displaystyle \max_{\Sigma^*}|u(T,x)|\,a_0T^{4/3}\xi^2r_0},
]

whence

[
\max_{\Sigma^*}|u(T,x)|
<
\frac{2^{16/3}M_1^{1/3}M_2^{1/3}\widetilde M_2(1-2\delta)^{2/3}}
{N_1^{2/3}\xi^3a_0T^{1/3}}.
\tag{6}
]

If at least one of the values (2\varepsilon_0) and (\varepsilon_0) does not belong to (\mathfrak M), we choose (2\varepsilon_0') and (\varepsilon_0''\in\mathfrak M), arbitrarily close to (2\varepsilon_0) and (\varepsilon_0), respectively, so that (2\varepsilon_0'-\varepsilon_0''=\varepsilon_0'''\gg\varepsilon_0). Carrying out arguments and calculations analogous to those carried out for the case when (2\varepsilon_0) and (\varepsilon_0\in\mathfrak M), we obtain that

[
\max_{\Sigma^*}|u(T,x)|=2\varepsilon_0\leq 2\varepsilon_0''
<
\frac{2^{16/3}M_1^{1/3}M_2^{1/3}\widetilde M_2(1-2\delta)^{2/3}}
{N_1^{2/3}\xi^3a_0T^{1/3}}.
\tag{6'}
]

Thus, in the case (\mathfrak D_1^0) we obtain inequality (3), and in the case (\mathfrak D_2^0), inequality (6). Since estimate (6) is the coarsest, it is suitable for all cases.

Theorem. Let, in the rectangular domain (G), the number (N_1) of essential components having limit points on (\Sigma^) satisfy condition (2). Then*

[
\max_{\Sigma^*}|u(T,x)|