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

\[ \left|\frac{\partial^2 u}{\partial x^2}\right| < \frac{\widetilde{M}_2(1-2\delta)^2 \max_{\overline{G}}|u|} {t\left[\left(\frac{1-2\delta}{2}\right)^2-\left(\frac{1-2x}{2}\right)^2\right]^2}. \]

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

\[ \left|\frac{\partial^2 u}{\partial x^2}\right| < \frac{2\widetilde{M}_2(1-2\delta)^2 \max_{\overline{G}}|u|} {T \xi^2(1-2\delta-\xi)^2}. \tag{4} \]

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

\[ \log_2 \frac{T\xi^2(1-2\delta-\xi)^2a_0r_0} {3\widetilde M_2(1-2\delta)^2\gamma_0} > \frac{4M_2}{\gamma_0^3}, \]

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

\[ \max_{\Sigma^*}|u(T,x)|<MN_1^{-2/3}\xi^{-3}, \]

where

\[ M= \frac{2^{16/3}M_1^{1/3}M_2^{1/3}\widetilde M_2(1-2\delta)^{2/3}} {a_0T^{1/3}} \]

depends only on the domain \(G\) and the coefficients of equation (1), and the positive number
\[ \xi<\frac{1-2\delta}{4}. \]

The author expresses deep gratitude to E. M. Landis for guidance and help.

Received
18 VII 1959

CITED LITERATURE

1. E. M. Landis, Uspekhi Mat. Nauk, 14, issue 1 (85), 21 (1959).
2. S. N. Bernstein, DAN, 18, No. 6, 385 (1938).
3. A. S. Kronrod, E. M. Landis, DAN, 58, No. 7, 1269 (1947).