E. M. LANDIS

ON THE DEPENDENCE BETWEEN THE NUMBER OF CHANGES OF SIGN OF A SOLUTION OF A PARABOLIC EQUATION AND THE GROWTH OF THE SOLUTION

(Presented by Academician I. G. Petrovsky, 8 VII 1958)

Let there be given the equation

\[ \frac{\partial u}{\partial t} = \sum_{i,k=1}^{n} a_{i,k}(t,x_1,\ldots,x_n) \frac{\partial^2 u}{\partial x_i\partial x_k} + \sum_{i=1}^{n} b_i(t,x_1,\ldots,x_n) \frac{\partial u}{\partial x_i} + c(t,x_1,\ldots,x_n)u \equiv Lu, \tag{1} \]

defined in a domain \(D_T\) lying in the cylinder \(0<t<T,\ |x|<1\), where \(x=(x_1,\ldots,x_n)\), and having limit points on each of the hyperplanes \(t=0\) and \(t=T\). Suppose that the coefficients of equation (1) satisfy the following conditions: the coefficients \(a_{i,k}\) are twice continuously differentiable, the coefficients \(b_i\) are continuously differentiable, and the inequalities

\[ |a_{i,k}|<M,\qquad \left|\frac{\partial a_{ik}}{\partial x_j}\right|<M,\qquad \left|\frac{\partial^2 a_{ik}}{\partial x_j\partial x_l}\right|<M,\qquad |b_i|<M,\qquad \left|\frac{\partial b_i}{\partial x_i}\right|<M, \]
\[ -M<C<0,\qquad i,k=1,\ldots,n,\quad j,l=1,\ldots,n; \tag{2} \]

are satisfied, and

\[ \sum_{i,k=1}^{n} a_{i,k}\xi_i\xi_k \bigg/ \sum_{i=1}^{n}\xi_i^2 > a>0. \tag{3} \]

Denote by \(S_T\) that part of the boundary of the domain \(D_T\) which is situated outside the hyperplanes \(t=0\) and \(t=T\). Let \(u(t,x)\) be a solution of equation (1), defined in \(D_T\), continuous in \(\overline{D_T}\), and vanishing on \(S_T\). Denote by \(D_T^+\) and \(D_T^-\) the sets of points \((t,x)\in D_T\) at which, respectively, \(u(t,x)>0\) and \(u(t,x)<0\). We shall call a component of the set \(D_T^+\) or \(D_T^-\) essential if it has limit points on each of the hyperplanes \(t=0\) and \(t=T\). The total number of all essential components of the sets \(D_T^+\) and \(D_T^-\) will be called the number of changes of sign of the function \(u(t,x)\) in \(D_T\).

Theorem. There exists a constant \(C\), depending on the constant \(M\) in inequalities (2) and on the constant \(a\) in inequality (3), such that for every solution \(u(t,x)\) of equation (1), defined in the domain \(D_T,\ T\ge 1\), continuous in \(\overline{D_T}\), and having in \(D_T\) \(N\) changes of sign, the inequality

\[ \max_x |u(T,x)| \big/ \max_x |u(0,x)| < e^{-T N^{2/n}/C}. \tag{4} \]

holds.

This theorem is adjacent to the theorems of note (¹), in which a similar result was formulated for solutions of elliptic equations. The proof of the theorem is based on lemmas likewise similar to the lemmas of note (¹).

Lemma 1. There exists a constant \(C_1\), depending on the constant \(M\) in inequality (2) and on the constant \(a\) in inequality (3), such that, under the condition

\[ \mu_{n+1}D_T \le T^{n/2+1}/C_1 \]

(where by \(\mu_k E\) we shall denote the \(k\)-dimensional measure of the set \(E\)), every solution \(u(t,x)\) of equation (1), positive in \(D_T\), continuous in \(\overline D_T\), and vanishing on \(\dot S_T\), satisfies the inequality

\[ \max_x u(T,x)/\max_x u(0,x)<{}^1\!/_{2}. \]

The idea of the proof of this lemma is as follows. One can find a subdomain \(g\) of the domain \(D_T\) having the following properties:

1) \(g\) is situated between the hyperplanes \(t=t_0\) and \(t=t_0+h^2\), \(0<h<1\), and has limit points on each of them. Denote by \(\sigma\) that part of the boundary of the domain \(g\) which is situated strictly between the hyperplanes \(t=t_0\) and \(t=t_0+h^2\), and by \(\gamma\) the remaining part of the boundary of \(g\). For each \(t_1\), \(t_0<t_1<t_0+h^2\), denote by \(\sigma_{t_1}\) the intersection of \(\sigma\) with the hyperplane \(t=t_1\).

2) \(\mu_{n+1}g<L_1h^{n+1}\) and \(\mu_n\gamma<L_2h^n\), where \(L_1\) and \(L_2\) are absolute constants.

3) \(u|_\sigma=\mathrm{const}=u_0\) and \(\partial u(t_1,x)/\partial n_{t_1}|_\sigma>0\) for almost all \(t_1\in[t_0,t_0+h^2]\), where \(\partial u(t_1,x)/\partial n_{t_1}\) is the derivative in the direction of the inner normal to \(\sigma_{t_1}\) in the hyperplane \(t=t_1\).

4)

\[ \int_\sigma \frac{\partial u(t,x)}{\partial n_t}\,d\sigma > C_1h^n\max_x u(T,x)/L_3, \]

where \(L_3\) is an absolute constant.

Applying to the left-hand side of the equality

\[ \int_g \left(\frac{\partial u}{\partial t}-Lu\right)\,dt\,dx_1\ldots dx_n=0 \]

Green’s formula, we obtain

\[ \int_{\gamma+\sigma} u\,dx_1\ldots dx_n - \int_\sigma \sum_{i,j} a_{ij}\frac{\partial u}{\partial x_j}\,dt\,dx_1\ldots dx_{i-1}dx_{i+1}\ldots dx_n+ \]

\[ + \int_\sigma \left( \sum_{i,j}\frac{\partial a_{ij}}{\partial x_i} - \sum_j b_j \right) u\,dt\,dx_1\ldots dx_{j-1}dx_{j+1}\ldots dx_n+ \]

\[ + \int_g \left( -\sum_{i,j}\frac{\partial^2 a_{ij}}{\partial x_i\partial x_j} + \sum_i\frac{\partial b_i}{\partial x_i} - C \right) u\,dt\,dx_1\ldots dx_n=0. \]

Applying inequalities (2), (3) and taking into account properties 1)—4) of the domain \(g\), we find

\[ aC_1h^n\max_x u(T,x)/L_3 < \bigl(L_1+(2n^2+2n+1)L_2Mh\bigr)h^n\max_x u(0,x). \]

It follows that, for sufficiently large \(C_1\), the inequality

\[ \max_x u(T,x)/\max_x u(0,x)<{}^1\!/_{2} \]

holds.

Lemma 2. Let \(P\) be the cylinder

\[ t^2+\sum_{i=2}^{n}x_i^2<h^2,\qquad 0<h<1. \]

Let \(\Gamma_1\) and \(\Gamma_2\) be \(n\)-dimensional manifolds lying inside the cylinder \(P\), with boundary on the boundary of \(P\), each of which separates in the cylinder \(P\) the points with sufficiently large positive coordinates \(x_1\) from the points

with sufficiently large, in absolute value, negative coordinates \(x_1\). Suppose that for the points \((t,x)\in \Gamma_1+\Gamma_2\) the inequality \(|x|<1\) is satisfied. Let \(G\) be the part of \(P\) lying between \(\Gamma_1\) and \(\Gamma_2\). Suppose that in \(G\) equation (1) is defined, satisfying in \(G\) the conditions (2) and (3). Let \(u(t,x)\) be a solution of this equation, defined in \(G\) and continuously differentiable in \(\bar G\). Suppose that \(u(t,x)\) satisfies the following conditions: 1) \(u|_{\Gamma_1}=u_0>0\), 2) \(u|_{\Gamma_2}=0\), 3) \(\partial u/\partial n|_{\Gamma_2}\leq 0\), where \(\partial/\partial n\) denotes differentiation along the inner normal, 4) \(u(t,x)\leq -u_0\) for \(x\in G\). Denote by \(G_1\) the set of points \((t,x)\in G\) for which \(u_0/2<u(t,x)<u_0\). Then

\[ \mu_{n+1}G_1>h^{n+1}/C_2, \]

where \(C_2\) is a constant depending on the constant \(M\) of inequalities (2) and on the constant \(a\) of inequality (3).

The idea of the proof of this lemma has common features with the idea of the proof of Lemma 1 of the present note and Lemma 2 of note (1). It is as follows. Put \(h_1^{\,n+1}/\mu_{n+1}G_1=C_2^*\). Let \(E_\tau\) be the level set \(\tau\) in \(G\). There exists such a \(u^*\), \(u_0/2<u^*<u_0\), that: a) the level set \(E_{u^*}\) contains no points \((t,x)\) where \(\operatorname{grad}u(t,x)=0\), and, consequently, consists of smooth \(n\)-dimensional manifolds; b) the inequality

\[ \int_{E_{u^*}}\left[\sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_i}\right)^2\right]^{1/2}d\sigma> \frac{C_2^*u_0h^{\,n-1}}{M_1}, \tag{5} \]

holds, where \(M_1\) is an absolute constant. Denote by \(G_{u^*}\) the set of points \((t,x)\in G\) for which \(u(t,x)<u^*\). Applying Green’s formula to the left-hand side of the equality

\[ \int_{G_{u^*}}\left(\frac{\partial u}{\partial t}-Lu\right)\,dt\,dx_1\ldots dx_n=0 \]

we obtain

\[ \int_{\bar\sigma_{u^*}-G_{u^*}}u\,dx_1\ldots dx_n- \int_{E_{u^*}+\Gamma_2}\sum_{i,j}a_{ij}\frac{\partial u}{\partial x_j}\,dt\,dx_1\ldots dx_{i-1}dx_{i+1}\ldots dx_n+ \]

\[ +\int_{E_{u^*}+\Gamma_2}\left(\sum_{i,j}\frac{\partial a_{ij}}{\partial x_i}-\sum_j b_j\right)u\,dt\,dx_1\ldots dx_{j-1}dx_{j+1}\ldots dx_n+ \]

\[ +\int_{G_{u^*}}\left(-\sum_{i,j}\frac{\partial^2 a_{ij}}{\partial x_i\partial x_j}+\sum_j\frac{\partial b_i}{\partial x_j}-c\right)u\,dt\,dx_1\ldots dx_n=0. \]

Hence, with the aid of inequalities (5), (2), (3) and conditions 3) and 4), we obtain

\[ aC_2^*u_0h^n/M_1<M\cdot M_2u_0h^h, \]

where \(M_2\) is an absolute constant. The last inequality gives us an estimate of \(C_2^*\) in terms of \(M\) and \(a\).

With the aid of Lemmas 1 and 2, Lemma 3 is proved, analogous to Lemma 3 of note (1).

Lemma 3. Suppose that in the domain \(D_1\) a solution \(u(t,x)\) of equation (1) is defined, continuous on \(\bar D_1\) and vanishing on the lateral surface \(S_1\) of the domain \(D_1\). Suppose that \(u(t,x)\) has \(N\) changes of sign in \(D_1\). Let \(g_1,\ldots,g_N\) be the essential components of the sets \(D_1^+\) and \(D_1^-\). Put

\[ m'=\max_x |u(0,x)|,\qquad m''=\max_x |u(1,x)|,\qquad m_i''=\max_{(T,x)\in \bar g_i}|u(T,x)|. \]

There exists a constant \(C_3\), depending on the constant \(M\) in inequality (2) and on the constant \(a\) in inequality (3), such that from the fact that

\[ m''/m' > 2^{-N^2/n/C_3}, \tag{6} \]

it follows that

\[ \min_i m_i'' > 2^{-N^2/n/[4(C_1\omega_n)^{2/n}]}, \tag{7} \]

where \(C_1\) is the constant of Lemma 1 and \(\omega_n\) is the volume of the unit \(n\)-dimensional ball.

From Lemmas 1 and 3 it follows:

Lemma 4. In the notation of the preceding lemma, the following assertion holds: there exists a constant \(C_4\), depending on the constant \(M\) in inequality (2) and on the constant \(a\) in inequality (3), such that

\[ m''/m' < 2^{-N^2/n/C_4}. \]

Proof. Suppose that inequality (6) is satisfied and, consequently, by Lemma 3, inequality (7) is satisfied. We have \(\sum_{i=1}^{N}\mu_{n+1}g_i \leqslant \omega_n\), where \(\omega_n\) is the volume of the unit \(n\)-dimensional ball. Therefore there exists an \(i_0\) such that

\[ \mu_{n+1}g_{i_0} \leqslant \omega_n/N. \tag{8} \]

Set \(N_1=[(N/2C_1\omega_n)^{2/n}]\). Suppose that \(N_1>3\) (in the case \(N_1 \leqslant 3\), the assertion of the lemma is easily obtained by constructing a barrier). Put \(t_k=k/N_1\). Let

\[ m_{i_0,k}=\max_{(t_k,x)\in g_{i_0}} |u(t_k,x)|. \]

Let these maxima be attained at points \(P_k\), \(k=0,1,\ldots,N_1\). Denote by \(H_k\), \(k=1,\ldots,N_1\), the layer \(t_{k-1}<t<t_k\), and by \(g_{i_0,k}\) the component of the intersection of \(g_{i_0}\) with this layer whose closure contains the point \(P_k\). By the maximum principle, \(g_{i_0,k}\) has boundary points on both boundaries of the layer \(H_k\). Consider those \(k\), \(k=1,2,\ldots,N_1\), for which

\[ \mu_n g_{i_0,k} \leqslant 1/C_1N_1^{\,n/2+1}. \tag{9} \]

The number of such \(k\) does not exceed \(N_1/2\), since otherwise we would have

\[ \mu_{n+1}g_{i_0} > \frac{N_1}{2}\,\frac{1}{C_1N_1^{\,n/2+1}} =\frac{1}{2C_1N_1^{\,n/2}} > \frac{\omega_n}{N}, \]

which contradicts (8). Consequently, there are at least

\[ N_1/2-1 > N^{2/n}/2(2C_1\omega_n)^{2/n} \]

distinct \(k\) for which the inequality opposite to (9) holds, i.e., such that Lemma 1 is applicable to \(g_{i_0,k}\). From this lemma and the maximum principle we obtain that for such \(k\) the inequality \(m_{i_0,k}<m_{i_0,k-1}/2\) holds. Hence

\[ m'' > 2^{N^{2/n}/[4(2C_1\omega_n)^{2/n}]}\min_i m_i'' \geqslant 2^{N^{2/n}/[4(2C_1\omega_n)^{2/n}]}m_{i_0} > \]

\[ >2^{N^{2/n}/[4(2C_1\omega_n)^{2/n}]} 2^{-N^{2/n}/[2(2C_1\omega_n)^{2/n}]} = 2^{-N^2/n/[4(2C_1\omega_n)^{2/n}]}. \]

It remains to put \(C_4=\max(C_3,\;4(2C_1\omega_n)^{2/n})\).

The theorem formulated above is a simple consequence of Lemma 4.

Received
4 VII 1958

References

1. E. M. Landis, DAN, 123, No. 4 (1958).