UDC 517.919

MATHEMATICS

A. V. IVANOV

A HARNACK INEQUALITY FOR GENERALIZED SOLUTIONS OF QUASILINEAR PARABOLIC EQUATIONS OF SECOND ORDER

(Presented by Academician V. I. Smirnov on 30 V 1966)

In the works \((^1,^2)\) J. Moser proposed a method for obtaining the Harnack inequality for generalized solutions (g.s.) from the classes \(\mathring W_2^1(\Omega)\) and \(\mathring W_2^1(Q)\) of the equations \((\mathcal A_{ij}u_{x_i})_{x_j}=0\) and \(u_t=(\mathcal A_{ij}u_{x_i})_{x_j}\). In \((^3)\) J. Serrin extended the result of \((^1)\) to g.s. of a certain class of quasilinear elliptic equations. Here we extend the Harnack inequality to g.s. of a certain class of quasilinear parabolic equations of second order (see \((^8)\)), and in the proof the approaches used in \((^{1-3})\)* are developed.

Let \(\Omega\) be some bounded \(n\)-dimensional domain in \(E_n\), \(Q\) the cylinder \(\Omega \times [t_1,t_2]\). By \(L_{p,p_1}(Q)\) \((p,p_1\geq 1)\) we denote the space of all measurable functions in \(Q\) for which the norm is finite

\[ \|u\|_{p,p_1,Q} = \left(\int_{t_1}^{t_2}\|u\|_{p,\Omega}^{p_1}\,dt\right)^{1/p_1} = \left(\int_{t_1}^{t_2}\left(\int_\Omega |u|^p\,dx\right)^{p_1/p}dt\right)^{1/p_1}. \tag{1} \]

For \(p_1=\infty\),

\[ \|u\|_{p,\infty,Q}=\operatorname{vrai\,max}_{t\in[t_1,t_2]}\|u\|_{p,\Omega}. \]

We introduce also the following notation. \(\mathring W_2^{1,0}(Q)\) is the closure of the set of all smooth and finite in \(Q\) functions in the norm \((\|u\|_{2,Q}^2+\|u_x\|_{2,Q}^2)^{1/2}\); \(\mathring W_2^{1,1}(Q)\) is the subset of all functions from \(\mathring W_2^{1,0}(Q)\) which vanish in some neighborhood of the lateral surface and the lower base of the cylinder \(Q\) and have the generalized derivative \(u_t\in L_2(Q)\); \(U_2^{1,0}(Q)\) is the space of all functions \(u(x,t)\) belonging to \(L_2(Q)\) and having generalized derivatives \(u_{x_i}\) from \(L_2(Q)\) and a finite \(\operatorname{vrai\,max}_{t\in[t_1,t_2]}\|u\|_{2,\Omega}\), with norm

\[ \langle\!\langle u\rangle\!\rangle_Q = \left(\operatorname{vrai\,max}_{t\in[t_1,t_2]}\|u\|_{2,\Omega}^2+\|u_x\|_{2,Q}^2\right)^{1/2}, \tag{2} \]

where

\[ u_x=(u_{x_1},\ldots,u_{x_n}),\qquad \|u_x\|_{2,Q} = \left\|\left(\sum_{i=1}^n u_{x_i}^2\right)^{1/2}\right\|_{2,Q}. \]

Denote by \(K_\rho(x_0)\) the \(n\)-dimensional ball of radius \(\rho\) \((\rho>0)\) with center at the point \(x_0\), and by \(Q_\rho(x_0,t_0)\) the cylinder \(K_\rho(x_0)\times[t_0-\rho^2,t_0]\).

Lemma 1. Let \(u(x,t)\in U_2^{1,0}(Q_\rho)\), and let the numbers \(l,l_1\) \((l,l_1\geq 1)\) satisfy the conditions

\[ 1/l+2/nl_1\geq 1/2, \]

\[ l\geq 2;\ l_1\geq 2\ \text{for } n\geq 3;\quad 2\leq l<\infty,\ l_1>2\ \text{for } n=2;\quad l\geq 2,\ l_1\geq 4 \quad \text{for } n=1. \tag{3} \]

* A certain generalization of the result of Moser’s work \((^2)\) was obtained in \((^4)\), where it was shown that the Harnack inequality will also hold for a somewhat broader class of g.s. of the equation \(u_t=(\mathcal A_{ij}u_{x_i})_{x_j}\) than \(\mathring W_2^1(Q)\).

Then the function \(u(x,t)\) belongs to the space \(L_{l,l_1}(Q_\rho)\) and satisfies the inequality

\[ \|u\|_{l,l_1,Q_\rho} \leq a_0\left(\operatorname{vrai\,max}_{t\in[t_1,t_2]}\|u\|_{2,K_\rho}\right)^{1-2/l_1} \left(\|u\|_{2,Q_\rho}+\|u_x\|_{2,Q_\rho}\right)^\varkappa \left(\|u\|_{2,Q_\rho}\right)^{2/l-\varkappa}, \tag{4} \]

where \(\varkappa=n/2-n/l\) (obviously, \(\varkappa\in[0,2/l_1]\)) and \(a_0\) depends only on \(n\) for \(n\ne2\) and on \(l\) for \(n=2\).

In the case \(1/l+2/nl_1>1/2\), inequality (4) implies the inequality

\[ \|u\|_{l,l_1,Q_\rho} \leq \varepsilon\langle u\rangle_{Q_\rho} + a_0'\,\varepsilon^{-\frac{1}{\varkappa-2/l_1}\frac{1+\varkappa-2/l_1}{\varkappa-2/l_1}} \|u\|_{2,Q_\rho}, \tag{5} \]

where \(\varepsilon\) is an arbitrary positive number.

Consider the equation

\[ \frac{\partial u}{\partial t} - \frac{\partial}{\partial x_i}\mathcal L_i(x,t,u,u_x) + \mathcal L_0(x,t,u,u_x)=0\ * , \tag{6} \]

where \(\mathcal L_i(x,t,u,p)\) and \(\mathcal L_0(x,t,u,p)\) \((p=(p_1,\ldots,p_n))\) are given functions of the variables \(x,t,u,p\), defined in the domain

\[ \mathcal R:\ x\in\Omega,\ t\in[0,T],\ -\infty<u<\infty,\ -\infty<p_k<\infty,\ k=1,\ldots,n, \]

measurable in \(x,t\) in the cylinder \(Q_T=\Omega\times[0,T]\) for fixed \(u\) and \(p\), and continuous in the variables \(u\) and \(p\) for fixed \(x\) and \(t\). Suppose that for all \((x,t,u,p)\in\mathcal R\) the inequalities

\[ \left[\sum_{i=1}^n \mathcal L_i^2(x,t,u,p)\right]^{1/2} \leq \mu |p|+\mathcal A|u|+\mathcal F, \]

\[ |\mathcal L_0(x,t,u,p)| \leq \mathcal B|p|+\mathcal C|u|+\mathcal G, \tag{7} \]

\[ \mathcal L_i(x,t,u,p)p_i \geq \nu |p|^2-\mathcal D|u|^2-\mathcal H, \]

hold, where \(\nu\) and \(\mu\) are positive constants; \(\mathcal A,\mathcal B,\mathcal C,\mathcal D,\mathcal F,\mathcal G,\mathcal H\) are nonnegative functions of the variables \(x,t\), belonging to the spaces

\[ \mathcal A,\mathcal B,\mathcal F\in L_{q,\infty}(Q_T), \qquad \mathcal C,\mathcal D,\mathcal G,\mathcal H\in L_{q/2,\infty}(Q_T), \qquad q>n\geq2. \tag{8} \]

A generalized solution from the class \(\mathring W^{1,0}_2(Q_T)\) of equation (6), whose coefficients satisfy conditions (7), (8), is a function

\[ u(x,t)\in U^{1,0}_2(Q_T), \]

satisfying, for almost all \(t_1,t_2\in[0,T]\) and all

\[ \Phi\in \mathring W^{1,1}_2(Q),\qquad Q=\Omega\times[t_1,t_2], \]

the integral identity

\[ \left.\int_\Omega u\Phi\,dx\right|_{t_1}^{t_2} + \int_{t_1}^{t_2}\int_\Omega \left[ -u\Phi_t + \mathcal L_i(x,t,u,u_x)\Phi_{x_i} + \mathcal L_0(x,t,u,u_x)\Phi \right]\,dx\,dt =0. \tag{9} \]

Denote by \(\Gamma_T\) the set of all points \((x,t)\) lying on the lateral surface or on the lower base of the cylinder \(Q_T\), and let

\[ \widehat Q_{2r} = \widehat Q_{2r}(x_0,t_0) = Q_{2r}(x_0,t_0+2r^2) = K_{2r}(x_0)\times[t_0-2r^2,t_0+2r^2]. \]

With the aid of the main lemma of [2] and Lemma 1 one proves

Theorem 1. Let \(u(x,t)\) be a generalized solution from the class \(U^{1,0}_2(Q_T)\) of equation (6), whose coefficients satisfy conditions (7), (8). Suppose that this generalized solution is nonnegative in the cylinder \(\widehat Q_{2r}(x_0,t_0)\), contained in \(Q_T\) and at a positive distance from \(\Gamma_T\). Then the function \(u(x,t)\) satisfies the inequality (Harnack)

\[ \operatorname{vrai\,max}_{Q_{r/4}^-} u \leq \operatorname{const}\left( \operatorname{vrai\,min}_{Q_{r/4}^+} u+k_r \right), \tag{10} \]

where

\[ Q_{r/4}^- = Q_{r/4}(x_0,t_0-\tfrac{7}{4}r^2), \qquad Q_{r/4}^+ = Q_{r/4}(x_0,t_0+2r^2), \]

\[ k_r = r^\delta\|\mathcal F\|_{q,\infty,\widehat Q_{2r}} + r^{2\delta}\|\mathcal G\|_{q/2,\infty,\widehat Q_{2r}} + r^\delta\left(\|\mathcal H\|_{q/2,\infty,\widehat Q_{2r}}\right)^{1/2}, \qquad \delta=1-\frac{n}{q}. \]

The constant in (10) depends only on

\[ n,\nu,\mu,q,\ r^\delta\|\mathcal A,\mathcal B\|_{q,\infty,\widehat Q_{2r}} \]

and

\[ r^{2\delta}\|\mathcal C,\mathcal D\|_{q/2,\infty,\widehat Q_{2r}}. \]

\(*\) By \(\dfrac{\partial}{\partial x_i}\mathcal L_i(x,t,u,u_x)\) is meant the total derivative of the function \(\mathcal L_i(x,t,u(x,t),u_x(x,t))\) with respect to the variable \(x_i\).

As a consequence of Harnack’s inequality one easily obtains an estimate of the Hölder constant of an arbitrary generalized solution from the class \(\mathscr{V}_{2}^{1,0}(Q_T)\) of equation (6), considered under conditions (7), (8).

In the case
\[ \mathscr{L}_i(x,t,u,p)=\sum_{j=1}^{n} L_{ij}p_j+A_i u+F_i,\qquad \mathscr{L}_0(x,t,u,p)=\sum_{j=1}^{n} B_jp_j+ \]
\[ +Cu+G \]
the estimate of the Hölder constant was first obtained by different methods by O. A. Ladyzhenskaya and N. N. Ural’tseva in \((^6)\), and (for \(A_i=B_i=F_i=C=G=0\)) by J. Nash in \((^5)\).

Theorem 2. Let \(u(x,t)\) be an arbitrary generalized solution from the class \(\mathscr{V}_{2}^{1,0}(Q_T)\) of equation (6), satisfying conditions (7), (8), and let \(Q'=\Omega'\times[\delta',T]\), \(Q''=\Omega''\times[\delta'',T]\), \(\overline{\Omega}'\subset\Omega''\subset\overline{\Omega}''\subset\Omega\), \(\delta'>\delta''>0\). Then the function \(u(x,t)\) has a finite \(\operatorname{vrai\,max}_{Q'}|u|\) and, for almost all \((x,t)\), \((\bar x,\bar t)\in Q'\), satisfies the inequality
\[ |u(x,t)-u(\bar x,\bar t)|\leqslant \operatorname{const}\left(\operatorname{vrai\,max}_{Q'}|u|+K\right) \left(|x-\bar x|^2+|t-\bar t|\right)^\gamma, \tag{11} \]
where \(\gamma\in(0,1]\),
\[ K=\|\mathscr{F}\|_{q,\infty,Q''}+\|\mathscr{G}\|_{q/2,\infty,Q''}+ \left(\|\mathscr{H}\|_{q/2,\infty,Q''}\right)^{1/2}, \]
and the constant in (11) depends only on \(n\), \(\nu\), \(\mu\), \(q\), the norms \(\|\mathscr{A},\mathscr{B}\|_{q,\infty,Q''}\), \(\|\mathscr{C},\mathscr{D}\|_{q/2,\infty,Q''}\), on the geometry of \(Q'\), and on the distance from \(Q'\) to the lateral surface and the lower base of the cylinder \(Q''\).

The functions \(\mathscr{A}, \mathscr{B},\ldots,\mathscr{H}\) may be regarded as elements of spaces \(L_{p,p_1}(Q_T)\) of general form \((p,p_1\geqslant1)\). In this case, for generalized solutions of the quasilinear equation (6) satisfying conditions (7), theorems (see \((^8)\)) analogous to Theorems 1 and 3–7 of paper \((^7a)\), concerning linear parabolic equations, are proved.

Remark added in proof.

1. The results of Theorems 1 and 2 remain valid if the functions \(\mathscr{A}, \mathscr{B},\ldots,\mathscr{H}\) are characterized in terms of belonging to spaces \(L_{p,p_1}(Q_T)\) of general form, i.e., if instead of (8) one imposes the condition
   \[ \mathscr{A},\mathscr{B},\mathscr{F}\in L_{2p,2p_1}(Q_T),\qquad \mathscr{C},\mathscr{D},\mathscr{G},\mathscr{H}\in L_{p,p_1}(Q_T) \]
   \[ n/p+2/p_1<2,\qquad p\geqslant1,\quad p_1\geqslant1,\quad n\geqslant1. \tag{8'} \]

2. In the case of bounded generalized solutions the results of these theorems remain valid if, instead of (7), one assumes that
   \[ \left[\sum_{i=1}^{n}\mathscr{L}_i^2\right]^{1/2}\leqslant \mu|p|+\mathscr{F},\qquad |\mathscr{L}_0|\leqslant\mu_1|p|^2+\mathscr{G},\qquad \mathscr{L}_i p_i\geqslant\nu|p|^2-\mathscr{H}, \tag{7'} \]
   where \(\mathscr{F}, \mathscr{G}\), and \(\mathscr{H}\) satisfy (8').

3. During the International Congress of Mathematicians in Moscow in 1966 it became known to us that, recently, a result analogous to Theorem 1 in the case of conditions (7), \((8^1)\) had been obtained by D. G. Aronson and J. Serrin (an announcement of their result appears in Notices of the Am. Math. Soc., 13 April 1966, p. 381), and, in the case of bounded generalized solutions under conditions \((7^1)\), \((8')\), also by N. S. Trudinger. In addition, for linear equations Harnack’s inequality has been extended by L. P. Kuptsov to a certain class of non-hyperbolic equations. These results have not yet been published.

CITED LITERATURE

\({}^1\) J. Moser, Comm. Pure and Appl. Math., 14, No. 3, 577 (1961).
\({}^2\) J. Moser, Comm. Pure and Appl. Math., 17, No. 1, 101 (1964).
\({}^3\) J. Serrin, Acta Math., 111, 247 (1964).
\({}^4\) D. G. Aronson, Ann. Polon. Math., 16, 3, 285 (1965).
\({}^5\) J. Nash, Am. J. Math., 80, No. 4, 931 (1958).
\({}^6\) O. A. Ladyzhenskaya, N. N. Ural’tseva, Izv. AN SSSR, ser. matem., No. 1, 5 (1962).
\({}^7\) A. V. Ivanov, O. A. Ladyzhenskaya et al., a) DAN, 168, No. 1 (1966); b) Tr. Matem. inst. im. V. A. Steklova AN SSSR, 92 (1966).
\({}^8\) Ivanov, Author’s abstract of Candidate dissertation, LSU, 1966.