MATHEMATICS

O. A. LADYZHENSKAYA and N. N. URAL’TSEVA

THE BOUNDARY VALUE PROBLEM FOR LINEAR AND QUASILINEAR PARABOLIC EQUATIONS

(Presented by Academician V. I. Smirnov on 17 III 1961)

Parabolic equations are considered in the cylinder \(Q_T=\Omega\times[0,T]\):

\[ u_t-\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i} \left(a_{ij}(x,t)u_{x_j}+b_i u+e_i\right) +\sum_{i=1}^{n}c_i u_{x_i}+du+f=0; \tag{1} \]

\[ u_t-\sum_{i=1}^{n}\frac{\partial}{\partial x_i} \left[a_i(x,t,u,u_{x_k})\right]+a(x,t,u,u_{x_k})=0 \tag{2} \]

with

\[ u\big|_{\Gamma}=\varphi(x,t) \tag{3} \]

where \(S\) is the boundary of the domain \(\Omega\), \(\Gamma=\{S\times[0,T]\}\cup\{\Omega,t=0\}\).

Numerous works are devoted to the study of linear parabolic equations. The sharpest result concerning estimates of their solutions was obtained in Nash’s paper \((^1)\). Namely, Nash gave an estimate in the norm \(C_{0,\alpha}\) of the fundamental solution of the equation

\[ u_t=\frac{\partial}{\partial x_i}\left(a_{ij}(x,t)u_{x_j}\right) \]

in terms of \(\max |a_{ij}|\) and the ellipticity constant. Among works on quasilinear parabolic equations with many independent variables, the strongest nonlinearities were considered in \((^2)\).

In the present work we have succeeded for the first time in proving solvability “in the large” of problem (2), (3) under such conditions on \(a_i\) and \(a\) which, in a certain sense, are necessary (see \((^3,^4)\)), and also in giving an estimate in the norm \(C_{0,\alpha}(Q_T)\)* of generalized solutions of the boundary value problem for linear parabolic equations under very small assumptions on the coefficients.

We mainly deal with a priori estimates of the solutions \(u\) and of their derivatives for equations (1) and (2) in Hölder norms \(C_{0,\alpha}\). It is known that generalized solutions of these equations satisfy a certain integral identity. In addition, as is not difficult to show, for solutions of (1) the inequality

\[ \frac{\partial}{\partial t} \int_{A_{k,\rho}(t)} |u-k|^2 \xi^2(x)\,dx +\nu \int_{A_{k,\rho}(t)} |\nabla u|^2 \xi^2\,dx \leq \]

\[ \leq \frac{\gamma}{\delta^2\rho^2} \int_{A_{k,\rho}(t)} |u-k|^2\,dx +\gamma\,\operatorname{mes}^{\,1-2/q} A_{k,\rho}(t) \tag{4} \]

holds, as well as the analogous inequality \((4')\) for \(B_{k,\rho}(t)\). Here and below \(K(\rho)\) denotes an arbitrary ball of radius \(\rho\); \(A_{k,\rho}(t)\) is the set

\[ \text{* The norm of } u(x,t) \text{ in } C_{0,\alpha}(Q_T) \text{ is defined by the equality} \]

\[ |u|_{C_{0,\alpha}(Q_T)} \equiv \max_{(x,t)\in Q_T}|u(x,t)| + \max_{(x,t),(x',t')\in Q_T} \frac{|u(x,t)-u(x',t')|} {\left(|x-x'|^2+|t-t'|\right)^{\alpha/2}}. \]

points of \(K(\rho)\cap\Omega\), where \(u(x,t)>k\); \(B_{k,\rho}(t)\) is the set of points of \(K(\rho)\cap\Omega\) where \(u(x,t)<k\). The function \(\xi(x)\) is continuously differentiable, nonnegative, equal to one in \(K(\rho-\sigma\rho)\) and to zero outside \(K(\rho)\). The constant

\[ \nu=2\min_{(x,t)\in Q_T}\frac{a_{ij}\xi_i\xi_j}{\sum_i \xi_i^2} \]

is here and below assumed to be positive; \(\gamma\) is determined by \(M=\operatorname{vrai}\max_{Q_T}|u|\), and also by

\[ \max_{Q_T}|a_{ij}|,\qquad \max_{0\le t\le T}\bigl(\|b_i,c_i,e_i\|_{L_q(\Omega)},\|d,f\|_{L_{q/2}(\Omega)}\bigr), \qquad q>n. \tag{5} \]

The number \(k\) in (4) is arbitrary for \(K(\rho)\subset\Omega\) and is larger than the maximum of \(u\) on the intersection of \(K(\rho)\) with \(S\), if the latter is nonempty.

With respect to the functions \(a_i(x,t,u,p_k)\) and \(a(x,t,u,p_k)\) we shall assume that the conditions

\[ a_i(x,t,u,p_k)p_i\ge \nu_0(|u|)p^2-\mu(|u|);\qquad p=\left(\sum_{i=1}^n p_i^2\right)^{1/2};\qquad \nu_0>0; \]

\[ \sum_i |a_i(x,t,u,p_k)|\,(p+1)|a(x,t,u,p_k)|\le \mu(|u|)(p^2+1);\qquad \mu>0 \tag{6} \]

are satisfied.

Then inequalities (4) and \((4')\) are also valid for solutions of equation (2), but not for all \(k\), rather for those for which \(\operatorname{mes} A_{k+\delta,\rho}(t)=0\) (respectively \(\operatorname{mes} B_{k-\delta,\rho}(t)=0\)), where \(\delta\) is a positive number determined by the constants \(\nu_0(M)\), \(\mu(M)\) from conditions (6).

Denote by \(\mathfrak{B}\) the set of functions satisfying inequalities (4) and \((4')\) for only the indicated \(k\) and for all \(t\in[0,T]\).

Theorem 1. The functions \(u(x,t)\) of the class \(\mathfrak{B}\) satisfy the Hölder condition

\[ |u(x,t)-u(x',t')|\le C d^{-\alpha}(x,t)\bigl[|x-x'|^2+|t-t'|\bigr]^{\alpha/2}. \tag{7} \]

with positive constants \(C\) and \(\alpha\), determined only by \(M,\gamma,\delta\), and \(\nu\). Here \(t\ge t'\), \(d(x,t)\) is the smaller of \(\sqrt{t}\) and the distance from \(x\) to \(S\).

We shall say that \(S\) satisfies condition \(A\) if there exist numbers \(a_0<1\) and \(A_0\) such that, for any sphere \(K(\rho)\) with center on \(S\) of radius \(\rho\le A_0\),

\[ \operatorname{mes}(K(\rho)\cap\Omega)\le a_0\,\operatorname{mes}K(\rho). \]

Denote by \(\Gamma\) the set of points lying on the lateral surface and the lower base of \(Q_T\).

Theorem 2. Let \(S\) satisfy condition \(A\). Then any function \(u\) from \(\mathfrak{B}\), whose boundary values on \(\Gamma\) satisfy the Hölder condition with some exponent \(\beta<0\), satisfies in \(\overline{Q}_T\) the Hölder condition

\[ |u(x,t)-u(x',t')|\le C\bigl[|x-x'|^2+|t-t'|\bigr]^{\alpha/2} \]

with certain positive constants \(\alpha\) and \(C\), determined by \(M,\gamma,\delta,\nu,a_0,A_0,|u|_{C_{0,\beta}(\Gamma)}\).

Theorems 1 and 2 give, for solutions of equations (1) and (2), an estimate of \(|u|_{C_{0,\alpha}}\) in terms of \(M=\operatorname{vrai}\max_{Q_T}|u|\). An estimate of \(M\) for solutions of equation (1) is given by the theorem:

Theorem 3. For a solution \(u\) of equation (1), \(M=\operatorname{vrai}\max_{Q_T}|u|\) is estimated in terms of \(\max_{\Gamma}|u|\) and the constants from (5).

With the aid of the a priori estimates guaranteed by Theorems 1–3, various existence theorems are established by known methods, for example, the following:

Theorem 4. Let the constants in (5) be finite and let \(S\) satisfy condition A. Then there exists a unique generalized solution \(u\) of problem (1), (3), belonging to \(C_{0,\alpha}(\overline{Q}_T)\), \(\alpha>0\), with \(u_{x_i}\in L_2(Q_T)\), if and only if \(u(x,0)\in C_{0,\beta}(\overline{\Omega})\), and \(u|_S=0\).

Suppose that \(a_i(x,t,u,p_k)\) and \(a(x,t,u,p_k)\), as functions of their arguments, belong to the classes \(C_{1,\alpha}\) and \(C_{0,\alpha}\), \(\alpha>0\). Suppose that the inequalities (6) are satisfied for them, so that the order of growth of \(a_i\) with respect to \(p\) is equal to 1, and the order of growth of \(a\) with respect to \(p\) is not greater than 2. Suppose that this order of \(a_i\), when \(a_i\) is differentiated with respect to \(p_k\), decreases by at least 1, and when differentiated with respect to \(u\) and \(x_k\) does not increase. Finally, suppose that the ellipticity condition is fulfilled in the form

\[ \nu(|u|)\sum_{i=1}^{n}\xi_i^2 \leq \frac{\partial a_i(x,t,u,p_k)}{\partial p_i}\,\xi_i\xi_j \leq \mu(|u|)\sum_{i=1}^{n}\xi_i^2,\qquad \nu>0. \]

Under these conditions the following is true.

Theorem 5. The norm of \(u_{x_i}\) in \(C_{0,\alpha}(Q'_T)\) and \(\max_{Q_T}|u_{x_i}|\) for solutions of equation (2) are estimated in terms of the data of the problem and \(M\), if \(S\in C_2\) and \(u|_\Gamma\in C_1\).

Here and below \(Q'_T=\Omega'\times[0,T]\), where \(\Omega'\) is an interior subdomain of \(\Omega\).

From this theorem and Friedman’s theorem \((^5)\) follows the possibility of estimating, for solutions \(u\) of equation (2), the norms of \(u_t\) and \(u_{x_i x_j}\) in \(C_{0,\alpha}(Q'_T)\) in terms of the data of the problem.

Theorem 6. Problem (2), (3) has a unique solution \(u\) from \(C_{0,\alpha}(\overline{Q}_T)\), with \(u_t, D_x^2u\) from \(C_{0,\alpha}(Q'_T)\) and \(\max |u_{x_i}|<\infty\), if \(u(x,0)\in C_{2,\alpha}(\Omega)\), \(u|_\Gamma\in C_1\), \(S\in C_2\), and if, with respect to \(a_i\) and \(a\), the conditions of Theorem 5 and the inequality

\[ \frac{\partial}{\partial u} \left[ -\frac{\partial a_i(x,t,u,0)}{\partial x_i} + a(x,t,u,0) \right] \geq \text{const}>-\infty \]

are satisfied for all \((x,t)\in \overline{Q}_T\) and arbitrary \(u\).

Theorems 1 and 2 are based on the following lemmas, valid for arbitrary functions from \(\mathfrak{B}\).

Lemma 1. Let \(K(\rho)\subset\Omega\) and \(\operatorname{mes} A_{k,\rho}(t_0)\leq a\chi_n\rho^n\), where \(\chi_n=\operatorname{mes}K(1)\). Then for any \(a<1\) there exist numbers \(\beta\) from \((0,1)\), \(h<1\), and \(\chi>0\), depending only on \(a\) and on the constants \(\gamma,\delta,\nu\) from (4), such that either

\[ H=\max |u(x,t)-k|\leq \rho^{\,1-n/q} \quad \text{for } x\in A_{k,\rho}(t),\quad t\in[t_0,t_0+\chi\rho^2], \]

or

\[ \max_{t\in[t_0,t_0+\chi\rho^2]} \operatorname{mes} A_{k+\beta H,\rho}(t) \leq h\chi_n\rho^n. \]

We shall denote by \(Q(\rho)\) cylinders of the form \(K(\rho)\times[t_0,t_0+\chi\rho^2]\). Take one of these cylinders and denote the oscillation of \(u\) in it by \(\omega\), the oscillation of \(u\) on the intersection of \(Q(\rho)\) with \(\Gamma\) (if it exists) by \(\omega_\Gamma\), and

\[ \max_{Q(\rho)}u=\mu_1,\qquad \min_{Q(\rho)}u=\mu_2. \]

Lemma 2. Let \(\omega\leq 4\omega_\Gamma\). Then for every \(\theta>0\) there exists an \(s\geq 2\) such that either

1) \(\omega\leq 2^s\rho^\varepsilon\), where \(\varepsilon=\min\{\beta,1-n/q\}\),

or

2) \(\operatorname{mes} A_{\mu_1-\omega/2^{s+1},\,\rho/4}(t)\leq \theta\rho^n\), for \(t\in[t_0+\tfrac{3}{4}\chi\rho^2,\ t_0+\chi\rho^2]\),

or

3) \(\operatorname{mes} B_{\mu_2+\omega/2^{s+1},\,\rho/4}(t)\leq \theta\rho^n\).

If \(t_0\leq 0\), then in 2) and 3) \(t\) may be taken from the interval \([0,t_0+\chi\rho^2]\).

Lemma 3. Let

\[ H=\max_{\substack{x\in A_{k,\rho}(t)\\ t\in[t_0,t_0+\Delta]}} |u(x,t)-k|\leq\delta, \qquad \Delta\geq\chi\rho^2, \]

Moreover, for \(K(\rho)\) intersecting \(S\), we consider only those \(k\) which exceed the greatest value of \(u\) on the intersection of \(K(\rho)\times [t_0,t_0+\Delta]\) with \(\Gamma\). Then for every \(\mu>0\) there exists a \(\theta>0\) such that if \(H\geqslant \rho^\varepsilon\), \(\varepsilon=\min\{\beta,1-n/q\}\), and \(\max\limits_{t\in [t_0,t_0+\Delta]}\operatorname{mes} A_{k,\rho}(t)\leqslant \theta \rho^n\), then \(\operatorname{mes} A_{k+H/2,\rho/2}(t)=0\) for \(t\in [t_0+\mu\rho^2,t_0+\Delta]\).

If \(t_0\leqslant 0\) and \(\operatorname{mes} A_{k,\rho}(0)=0\), then the latter assertion is valid for \(t\in [0,t_0+\Delta]\).

Assertions analogous to Lemmas 1 and 3 are also true for the sets \(B_{k,\rho}(t)\). We have formulated all the results under the assumption \(n\geqslant 2\). For \(n=1\) many of them take a simpler form. For problem (2), (3) the existence of a classical solution \(u(x,t)\) with \(u_t\) and \(u_{xx}\) in \(C_{0,\alpha}(\overline Q_T)\) has been proved.

Leningrad State University
named after A. A. Zhdanov

Received
13 III 1961

References

1. J. Nash, Am. J. Math., 80, No. 4, 931 (1958).
2. O. A. Ladyzhenskaya, Tr. Mosk. matem. obshch., vol. 7, 149 (1958).
3. O. A. Ladyzhenskaya, N. N. Ural’tseva, UMN, 16, issue 1 (1961).
4. O. A. Ladyzhenskaya, N. N. Ural’tseva, DAN, 135, No. 6, 1330 (1960).
5. A. Friedman, J. Math. and Mech., 7, No. 5, 771 (1958).