MATHEMATICS

S. N. KRUZHKOV

ON AN A PRIORI ESTIMATE OF SOLUTIONS OF LINEAR PARABOLIC EQUATIONS AND THE SOLUTION OF BOUNDARY-VALUE PROBLEMS FOR A CERTAIN CLASS OF QUASILINEAR PARABOLIC EQUATIONS

(Presented by Academician I. G. Petrovskii, 2 II 1961)

In the present note, J. Nash’s result \((^{1})\) on the continuity of solutions of parabolic equations is generalized. The a priori estimate obtained below for the modulus of continuity of solutions of equations of the form

\[ \frac{\partial u}{\partial t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( a_{ij}(t,x)\frac{\partial u}{\partial x_j} \right) + \sum_{i=1}^{n} b_i(t,x)\frac{\partial u}{\partial x_i} + f(t,x), \qquad x=(x_1,\ldots,x_n), \tag{1} \]

is then applied to prove the existence of a solution of the first boundary-value problem and of the Cauchy problem for a certain class of quasilinear parabolic equations; the proof of the existence theorems is carried out according to the scheme proposed in the work of O. A. Oleinik \((^{2})\).

With respect to equation (1) we shall always assume that its coefficients are sufficiently smooth, \(a_{ij}=a_{ji}\), and that the following inequalities are satisfied:

\[ \mu_1 \sum_{i=1}^{n}\xi_i^2 \le \sum_{i,j=1}^{n} a_{ij}\xi_i\xi_j \le \mu_2 \sum_{i=1}^{n}\xi_i^2, \qquad 0<\mu_1\le \mu_2; \]

\[ |b_i|\le B,\quad i=1,\ldots,n; \qquad |f|\le N. \tag{2} \]

Let \(\Omega\) denote a certain domain in the space \((x_1,\ldots,x_n)\); by \(\Omega^\delta\subset \Omega\) we denote the largest domain whose distance to the boundary of \(\Omega\) is equal to \(\delta\); by \(Q^\delta\) the cylinder \(\{\Omega^\delta\times[0,T]\}\); and by \(A\), \(\alpha\), and \(\beta\) we shall denote constants depending only on \(\mu_1,\mu_2\), and \(n\).

Theorem 1. Let \(u(t,x)\) be a solution of equation (1) in the cylinder \(Q\{\Omega\times[0,T]\}\); \(|u|\le M\). Then for \((t_1,x_1)\), \((t_2,x_2)\in Q^\delta\), \(0<t_1\le t_2\), \(0<\delta\le1\), the inequality

\[ |u(t_2,x_2)-u(t_1,x_1)| \le A\max\left[ \frac{M+N}{\delta^\alpha}, (M+N)B^\alpha, \frac{M}{\min(\sqrt{t_1},1)} \right]|x_2-x_1|^\alpha + \]

\[ + A\max\left[ \frac{M+N}{\delta^{2\beta}}, (M+N)B^{2\beta}, \frac{M}{\min(\sqrt{t_1},1)} \right](t_2-t_1)^\beta \tag{3} \]

holds for some \(\alpha\in(0,\tfrac12]\), \(\beta\in(0,\tfrac14]\).

Proof. It is sufficient to prove the theorem for the case when \(\Omega\) is the ball \(K\{|x|<R\}\). First we establish inequality (3) for solutions of equation (1) with \(b_i\equiv0\), \(i=1,\ldots,n\):

\[ \frac{\partial u}{\partial t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( a_{ij}(t,x)\frac{\partial u}{\partial x_j} \right) + f(t,x), \tag{4} \]

whose coefficients may be assumed given in the whole strip \(0\le t\le T\).

The following estimate is valid, and will be used below: if \(v(t,x)\) is a solution of equation (4) in the strip \(0 \leq t \leq T\), \(|v|\leq M\), then for any two points \((t_1,x_1)\), \((t_2,x_2)\), \(0<t_1\leq t_2\leq T\), the inequality
\[ |v(t_2,x_2)-v(t_1,x_1)|\leq A(M+Nt_1)(|x_2-x_1|/\sqrt{t_1})^\alpha \]
\[ {}+A\{M(1+t_1^\beta)+N(t_1+t_1^\beta)\}[(t_2-t_1)/t_1]^\beta \tag{5} \]
holds for some \(\alpha\in(0,1]\) and \(\beta=\alpha/2(1+\alpha)\).

The proof of this inequality follows easily from the result of J. Nash \({}^{(1)}\), if one uses the representation of the solution of equation (4) with zero initial conditions by means of the fundamental solution.

Let \((t_0,x)\), \((t_0,x+\Delta x)\in Q^\delta\), \(t_0>0\). Put
\[ \tau=\delta^{2/(1+\alpha)}|\Delta x|^{2\alpha/(1+\alpha)}, \]
where \(\alpha\) is the a priori constant from estimate (5).

Assume first that \(|\Delta x|\) is so small that \(\tau\leq \min(t_0,1)\). Define the function \(u_1(t,x)\) for \(t_0-\tau\leq t\leq T\) as the solution of the Cauchy problem for equation (4) with initial condition \(u_1(t_0-\tau,x)=u^0(x)\), where \(u^0(x)\) coincides with \(u(t_0-\tau,x)\) for \(|x|\leq R\), is continuous in the whole space, and \(|u^0(x)|\leq M\). Obviously,
\[ |u_1(t,x)|\leq M+N(t-t_0+\tau) \]
and for \(t\leq t_0\),
\[ |u_1(t,x)|\leq M+N\tau\leq M+N. \]

The function \(u_2(t,x)\equiv u(t,x)-u_1(t,x)\) is a solution of the homogeneous equation in the cylinder \(\{K\times[t_0-\tau,T]\}\), and \(u_2(t_0-\tau,x)=0\); moreover, for \(t_0-\tau\leq t\leq t_0\),
\[ |u_2(t,x)|\leq 2M+N\leq 2(M+N). \]
Using estimate (36) from \({}^{(1)}\), we obtain that for \(|x|\leq R-\delta\)
\[ |u_2(t_0,x)|\leq A(M+N)\sqrt{\tau}/\delta . \tag{6} \]

Applying inequality (5) to the function \(u_1(t,x)\) and taking (6) into account, we have
\[ |u(t_0,x+\Delta x)-u(t_0,x)|\leq \]
\[ \leq |u_1(t_0,x+\Delta x)-u_1(t_0,x)|+|u_2(t_0,x+\Delta x)|+|u_2(t_0,x)| \]
\[ \leq A(M+N+N\tau)(|\Delta x|/\sqrt{\tau})^\alpha +A(M+N)\sqrt{\tau}/\delta \]
\[ \leq A(M+N)\bigl(|\Delta x|^\alpha/\tau^{\alpha/2}+\sqrt{\tau}/\delta\bigr). \]
If instead of \(\tau\) we substitute its expression in terms of \(|\Delta x|\) and \(\delta\), we obtain
\[ |u(t_0,x+\Delta x)-u(t_0,x)| \leq A(M+N)(|\Delta x|/\delta)^{\alpha/(1+\alpha)}. \]
This estimate was obtained under the condition that \(\tau\leq\min(t_0,1)\), i.e.
\[ |\Delta x|^{\alpha/(1+\alpha)}\leq \min(\sqrt{t_0},1)/\delta^{1/(1+\alpha)}. \]
But if
\[ |\Delta x|^{\alpha/(1+\alpha)}>\min(\sqrt{t_0},1)/\delta^{1/(1+\alpha)}, \]
then, taking \(\delta\leq1\),
\[ \frac{|u(t_0,x+\Delta x)-u(t_0,x)|}{|\Delta x|^{\alpha/(1+\alpha)}} < \frac{2M}{\min(\sqrt{t_0},1)}\delta^{1/(1+\alpha)} \leq \frac{2M}{\min(\sqrt{t_0},1)} . \]
Thus, for \((t_0,x)\) and \((t_0,x+\Delta x)\in Q^\delta\),
\[ |u(t_0,x+\Delta x)-u(t_0,x)| \leq A\max\left[ \frac{M+N}{\delta^{\alpha/(1+\alpha)}}, \frac{M}{\min(\sqrt{t_0},1)} \right]|\Delta x|^{\alpha/(1+\alpha)} . \]

The second part of inequality (3), concerning continuity with respect to time of solutions of equation (4), is proved analogously. We note that a similar method was used in the work of J. Nash \({}^{(1)}\).

Let now \(u(t,x)\) be a solution of equation (1) in the cylinder \(Q\). The function \(u(t,x)\) can be regarded \({}^{(3)}\) as a solution of the equation
\[ \frac{\partial u}{\partial t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( a_{ij}(t,x)\frac{\partial u}{\partial x_j} \right) + \sum_{i=1}^{n} \frac{\partial}{\partial y} \left( b_i(t,x)y\frac{\partial u}{\partial x_i} \right) + \]
\[ {}+ \sum_{i=1}^{n} \frac{\partial}{\partial x_i} \left( b_i(t,x)y\frac{\partial u}{\partial y} \right) + \frac{\partial^2 u}{\partial y^2} + f(t,x) \tag{7} \]

in the cylinder \(Q'\{\Omega'\times[0,T]\}\), where \(\Omega'\) is the cylinder \(\{\Omega\times(0,s)\}\) in the space \((x_1,\ldots,x_n,y)\).

It is not difficult to see that for \(s=\min(\mu_1,1/n)/2B\) equation (7) is uniformly parabolic in \(Q'\), and the lower and upper bounds of the eigenvalues of the matrix of coefficients at the second derivatives depend only on \(\mu_1\) and \(\mu_2\). Since (7) is an equation of the form (4), by what has been proved, for \(u(t,x)=\widetilde u(t,x,y)\), in the cylinder \(Q^\delta\), for \(\delta\le s/4\), the inequality holds
\[ |u(t_2,x_2)-u(t_1,x_1)| \le A\max\left(\frac{M+N}{\delta^\alpha},\frac{M}{\min(\sqrt{t_1},1)}\right)|x_2-x_1|^\alpha + \]
\[ {}+ A\max\left(\frac{M+N}{\delta^{2\beta}},\frac{M}{\min(\sqrt{t_1},1)}\right)(t_2-t_1)^\beta . \tag{8} \]
Taking into account that for \(\delta>s/4\), \(Q^{s/4}\supset Q^\delta\), from (8) we obtain (3).

Remark. Denote by \(Q_{\delta_0/2}\) the cylinder \(\{\Omega^{\delta_0/2}\times[\delta_0/2,T]\}\subset Q\). In what follows inequality (3) will be applied in some cylinder \(Q_{\delta_0/2}\) for large \(B\) and fixed \(\mu_1,\mu_2,M,N,\delta_0\), and \(T\). But there exists a \(B^*(\mu_1,\mu_2,\delta_0)\) such that for \(B\ge B^*\) inequality (3) takes the form
\[ |u(t_2,x_2)-u(t_1,x_1)| \le A(M+N)\bigl[B^\alpha |x_2-x_1|^\alpha+B^{2\beta}(t_2-t_1)^\beta\bigr]. \tag{9} \]

Theorem 2. Let \(\overline Q\) be the cylinder \(\{\overline\Omega\times[0,T]\}\), whose base \(\overline\Omega\) has a three-times continuously differentiable boundary; let \(S\) be the lateral surface of the cylinder \(\overline Q\). Let a function \(\varphi(t,x)\in C^{2+\nu}\), \(|\varphi(t,x)|\le M_0\), be prescribed in \(\overline Q\). Then in the cylinder \(Q\) there exists a unique solution \(u(t,x)\) of the problem
\[ \frac{\partial u}{\partial t} = \sum_{i,j=1}^n \frac{\partial}{\partial x_i} \left(a_{ij}(t,x,u)\frac{\partial u}{\partial x_j}\right) + \]
\[ {}+ \sum_{i=1}^n b_i(t,x,u,u_x)\frac{\partial u}{\partial x_i} + c(t,x,u,u_x), \qquad u_x=(u_{x_1},\ldots,u_{x_n}); \tag{10} \]
\[ u\big|_S=\varphi(t,x)\big|_S,\qquad u(0,x)=\varphi(0,x), \tag{11} \]
if the following conditions are satisfied:

1) for \((t,x)\in Q\) and arbitrary \(u,u_x\),
\[ b_i\in C^{1+\nu},\qquad c\in C^{1+\nu}\quad \text{(locally)},\qquad c_u\le c_0,\quad |c(t,x,0,0)|\le c_1; \]

2) for \((t,x)\in \overline Q\), \(|u|\le M(M_0,c_0,c_1,T)\),
\[ a_{ij}=a_{ji},\qquad a_{ij}\in C^{2+\nu}, \]
\[ \mu_1\sum_{i=1}^n \xi_i^2 \le \sum_{i,j=1}^n a_{ij}\xi_i\xi_j \le \mu_2\sum_{i=1}^n \xi_i^2; \]

3) for \((t,x)\in \overline Q\), \(|u|\le M\), \(|\operatorname{grad}_x u|\ge P_0\ge1\),
\[ |b_i|\le B_0|\operatorname{grad}_x u|^\gamma,\qquad 0\le \gamma<1, \]
\[ |(b_i)_{x_k}|+|(b_i)_u|\le B_1|\operatorname{grad}_x u|,\qquad |(b_i)_{u_{x_k}}|\le B_2, \]
\[ |c|+|c_{x_k}|+|c_u|+|c_{u_{x_k}}|\le N, \]
and on the boundary of the lower base of the cylinder \(\overline Q\) the compatibility condition is satisfied
\[ \frac{\partial\varphi}{\partial t} = \sum_{i,j=1}^n \frac{\partial}{\partial x_i} \left(a_{ij}(t,x,\varphi)\frac{\partial\varphi}{\partial x_j}\right) + \sum_{i=1}^n b_i(t,x,\varphi,\varphi_x)\frac{\partial\varphi}{\partial x_i} + c(t,x,\varphi,\varphi_x). \]

The solution \(u(t,x)\) possesses first and second derivatives with respect to \(x_k\) and a first derivative with respect to \(t\), satisfying a Hölder condition in \(Q\).

In the case when \(b_i\equiv0\) \((i=1,\ldots,n)\), Theorem 2 was proved in \((^2)\). Theorem 2 is proved according to the same scheme as the corresponding result in \((^2)\), after an a priori estimate for \(|\operatorname{grad}_x u|\) has been obtained. The proof of this estimate is given below.

Let \(u(t,x)\) be a solution of problem (10), (11) in the cylinder \(\overline Q\); \(|u|\le M\). First, just as in \((^2)\), it is proved that \(|\operatorname{grad}_x u|\)

can be estimated in terms only of the data of problem (10), (11) in a certain boundary region \(\widetilde Q\setminus Q_{\delta_0}\{\Omega_{\delta_0}\times[\delta_0,T]\}\).

Let
\[ P=\max_{\overline Q}|\operatorname{grad}_x u| \]
and suppose that \(P\) is attained in \(Q_{\delta_0}\) at some point \((t_0,x_0)\). We may assume that \(P\ge P_0\) and \(B_0P^\gamma\ge B^*\) (see (9)); evidently,
\[ |b_i(t,x,u,u_x)|\le B_0P^\gamma,\qquad i=1,\ldots,n. \]
Define \(\rho\) and \(\tau\) from the relations
\[ \omega/2=A(M+N)B_0^\alpha P^{\alpha\gamma}\rho^\alpha=A_1P^{\alpha\gamma}\rho^\alpha, \]
\[ \omega/2=A(M+N)B_0^{2\beta}P^{2\beta\gamma}\tau^\beta=A_2P^{2\beta\gamma}\tau^\beta, \tag{12} \]
where \(\alpha,\beta\) and \(A\) are the constants from estimate (9), while the constant \(\omega\) will be chosen below (by \(A_i\) we denote constants depending only on the data of problem (10), (11)). First, let
\[ \omega\le \min\bigl[2A_1(\delta_0/2)^\alpha,\;2A_2(\delta_0/2)^\beta\bigr]=\widetilde\omega; \]
then \(\rho\le\delta_0/2,\ \tau\le\delta_0/2\). Denote by \(\widetilde Q\) the cylinder \(\{|x-x_0|\le \rho/2\times[t_0-\tau,t_0]\}\); evidently, \(\widetilde Q\subset Q_{\delta_0/2}\). In the cylinder \(\widetilde Q\) one may apply inequality (9) (here \(B=B_0P^\gamma\)); therefore in \(\widetilde Q\)
\[ |u(t_2,x_2)-u(t_1,x_1)|\le \]
\[ \le A(M+N)\bigl(B_0^\alpha P^{\alpha\gamma}\rho^\alpha+B_0^{2\beta}P^{2\beta\gamma}\tau^\beta\bigr) =\omega/2+\omega/2=\omega. \]

As in paper \((^2)\), when estimating \(|\operatorname{grad}_x u|\), in the cylinder \(\widetilde Q\) we make the substitution \(u=\varphi(v)\), under the condition that
\[ \omega\le\omega_0 \]
(\(\omega_0\) is determined by the data of problem (10), (11);
\[ 0<a_1(\omega_0)\le\varphi'(v)\le a_2(\omega_0) \])
and consider the function
\[ w(t,x)=\eta(t)\xi^2(x)q^2, \]
where
\[ q=|\operatorname{grad}_x v|; \]
\(\xi(x)\) and \(\eta(t)\) are smooth functions, with
\[ 0\le\xi(x),\eta(t)\le1,\quad \xi(x)=0\ \text{for } |x-x_0|\ge\rho/2,\quad \eta(t_0-\tau)=0, \]
\[ \xi(x_0)=\eta(t_0)=1,\quad \sum_{i=1}^n|\xi_{x_i}|^2+\sum_{i,j=1}^n|\xi_{x_ix_j}|\le A/\rho^2,\quad |\eta'|\le A/\tau. \]

It can be shown that at the point \((\bar t,\bar x)\) of the maximum of the function \(w(t,x)\)
\[ \bar w=w(\bar t,\bar x)\le \max[1,A_3(1/\rho^2+1/\tau)]. \]
Finally choosing
\[ \omega=\min(\omega_0,\widetilde\omega) \]
and substituting instead of \(\rho\) and \(\tau\) their expressions from (12), we have
\[ \bar w\le A_4P^{2\gamma} =A_4[\varphi'(v)]^{2\gamma}w^\gamma(t_0,x_0)\le A_5\bar w^\gamma, \]
i.e.
\[ \bar w\le A_5^{1/(1-\gamma)}=A_6. \]
Thus, everywhere in \(\widetilde Q\),
\[ w(t,x)\le\max(1,A_6)=A_7. \]
But
\[ w(t_0,x_0)=q^2=P^2/[\varphi'(v)]^2\le A_7; \]
hence
\[ P\le A_8. \]

A similar method can be applied to the construction of a solution of the Cauchy problem, and also to the construction of a solution of the Dirichlet problem for quasilinear elliptic equations; for elliptic equations, an approach connected with a generalization of the result of Giorgi \((^4)\) is given in paper \((^5)\).

The author expresses deep gratitude to O. A. Oleinik for valuable advice and attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
1 II 1961

CITED LITERATURE

\(^1\) J. Nash, Am. J. Math., 80, No. 4 (1958); translated collection, Matematika, 4, No. 1, 1960.
\(^2\) O. A. Oleinik, DAN, 138, No. 1 (1961).
\(^3\) O. A. Oleinik, S. N. Kruzhkov, UMN, 15, issue 5 (95), 203 (1960).
\(^4\) E. de Giorgi, Mem. Accad. Sci. Torino, ser. 3, 3, parte 1, 25 (1957).
\(^5\) O. A. Ladyzhenskaya, N. N. Ural’tseva, DAN, 135, No. 6 (1960).