UDC 517.946

MATHEMATICS

A. V. IVANOV, O. A. LADYZHENSKAYA, A. L. TRESKUNOV, N. N. URAL’TSEVA

SOME PROPERTIES OF GENERALIZED SOLUTIONS OF SECOND-ORDER PARABOLIC EQUATIONS

(Presented by Academician V. I. Smirnov on 9 VII 1965)

The paper investigates generalized solutions (g.s.) of linear and quasilinear uniformly parabolic second-order equations, to which the papers \((^{1-6})\) are devoted. Here, to characterize the coefficients of the equation, a continuous scale of norms of the spaces \(L_{p,p'}(Q_T)\) is used, i.e. the norm

\[ \|u\|_{p,p_1,Q_T} = \left( \int_0^T \left( \int_\Omega |u|^p dx \right)^{p_1/p} dt \right)^{1/p_1}, \tag{1} \]

where \(\Omega\) is some bounded domain contained in the Euclidean space \(E_n\), and \(Q_T=\Omega\times[0,T]\).

Let \(Q=\Omega\times[t_1,t_2]\). By \(V_2^{1,0}(Q)\) \(\bigl(\dot V_2^{1,0}(Q)\bigr)\) we denote the closure of all functions smooth in \(\overline Q\) (all smooth functions equal to zero near the lateral surface of \(Q\)) in the norm

\[ u\|_{V_2^{1,0}(Q)} = \left( \max_{t\in[t_1,t_2]}\int_\Omega u^2(x,t)\,dx + \iint_Q |\operatorname{grad}u|^2\,dx\,dt \right)^{1/2}. \tag{2} \]

Denote by \(K_\rho=K_\rho(x_0)\) the ball of radius \(\rho\) with center at the point \(x_0\) contained in \(E_n\); by \(Q_{\rho,\tau}=Q_{\rho,\tau}(x_0,t_0)\) the cylinder \(K_\rho(x_0)\times[t_0-\tau,t_0]\), and let \(v_+=\max(v,0)\).

Lemma 1. Let the numbers \(r>0\), \(k'\ge 0\) be fixed, and let the function \(v(x,t)\) belong to \(V_2^{1,0}(Q_{r,r^2})\) and, for all \(k\ge k'\), \(\sigma\in(0,1)\), \(\theta\in(0,1)\), \(\rho\in[r/2,r]\), \(\tau\in[(r/2)^2,r^2]\), satisfy the inequality

\[ \max_{t\in[t_0-\tau,t_0]} r^{-n} \int_{(K_{\rho-\sigma\rho},\,v(t)>k)} (v-k)^2\,dx + r^{-n} \iint_{(Q_{\rho-\sigma\rho,\tau-\theta\tau},\,v>k)} |\operatorname{grad}v|^2\,dx\,dt \le \]

\[ \le \gamma_0(\sigma^{-2}+\theta^{-1})r^{-(n+2)} \iint_{(Q_{\rho,\tau},\,v>k)} (v-k)^2\,dx\,dt + \]

\[ + \gamma_1 k^\delta \left\{ r^{-2} \int_{t_0-\tau}^{t_0} \left[ \frac{\operatorname{mes}(K_\rho,\,v(t)>k)} {\operatorname{mes}K_r} \right]^{\alpha-2/n} dt \right\}^{\beta}, \tag{3} \]

where \(\gamma_0\ge0,\ \gamma_1\ge0,\ 0\le\delta\le2,\ 1\le\alpha\le\infty,\ \beta\ge0,\ \alpha\beta=1+\varepsilon,\ \varepsilon>-1\).*

* We agree that for \(\alpha=\infty\) the expression

\[ \left\{ r^{-2} \int_{t_0-\tau}^{t_0} \left[ \operatorname{mes}(K_\rho,\,v(t)>k)\operatorname{mes}^{-1}K_r \right]^{\alpha-2/n} dt \right\}^{(1+\varepsilon)/\alpha} \]

is understood as the limit of the written expression as \(\alpha\to\infty\), i.e. as

\[ \max_{t\in[t_0-\tau,t_0]} \left[ \operatorname{mes}(K_\rho,v(t)>k)\cdot\operatorname{mes}^{-1}K_r \right]^{1+\varepsilon}. \]

1) Let \(-1<\varepsilon<0,\ \delta=0\). Then
\[ r^{-(n+2)} \operatorname{mes}(Q_{r/2,r^2/2},\, v>h) \le c_1 h^{-2(n+2)/n|\varepsilon|} \left(r^{-(n+2)/2}\|v_+\|_{2,2,Q_{r,r^2}}+1\right)^{2(n+2)/n|\varepsilon|}, \tag{4} \]
\[ h\ge h_0=c_0^{(1)}\left(r^{-(n+2)/2}\|v_+\|_{2,2,Q_{r,r^2}}+1\right). \]

2) Let \(\varepsilon=0,\ 0\le\delta\le2\), and let \(p\) be any number greater than \(1\). There exists a number \(d\), depending only on \(n\), such that for \(\gamma_1\le de^{-p}\) the inequality holds
\[ r^{-(n+2)} \operatorname{mes}(Q_{r/2,r^2/2},\, v>h) \le c_2 h^{-p}\left(r^{-(n+2)/2}\|v_+\|_{2,2,Q_{r,r^2}}+1\right)^p, \tag{5} \]
\[ h\ge h_0=c_0^{(2)}\left(r^{-(n+2)/2}\|v_+\|_{2,2,Q_{r,r^2}}+1\right). \]

3) Let \(\varepsilon=0,\ \delta=0\). Then
\[ r^{-(n+2)} \operatorname{mes}(Q_{r/2,r^2/2},\, v>h) \le c_3 \exp\left\{-c_4\left(r^{-(n+2)/2}\|v_+\|_{2,2,Q_{r,r^2}}+1\right)^{-1}h\right\}, \tag{6} \]
\[ h\ge h_0=c_0^{(3)}\left(r^{-(n+2)/2}\|v_+\|_{2,2,Q_{r,r^2}}+1\right). \]

4) Let \(\varepsilon>0,\ 0\le\delta\le2\). Then
\[ \operatorname*{vrai\,max}_{Q_{r/2,r^2/2}} v \le c_5\left(r^{-(n+2)/2}\|v_+\|_{2,2,Q_{r,r^2}}+1\right). \tag{7} \]

In inequalities (4)—(7), the constants \(c_0^{(i)}\) \((i\ne2)\) and \(c_j\) \((j\ne2)\) depend only on \(n,\gamma_0,\gamma_1,k',\alpha\), and \(\beta\), while the constants \(c_0^{(2)}\), \(c_2\) depend on the same quantities and on the number \(p\).

For \(\alpha=1+2/n\), Lemma 1 was proved in \((^6)\).

Consider in \(Q_T\) the linear equation
\[ \frac{\partial u}{\partial t} - \frac{\partial}{\partial x_i} \left(A_{ij}\frac{\partial u}{\partial x_j}+A_i u\right) + B_i\frac{\partial u}{\partial x_i} + Cu + \frac{\partial}{\partial x_i}F_i + G =0 \tag{8} \]
under the condition that
\[ \nu\sum_{i=1}^n \xi_i^2 \le A_{ij}(x,t)\xi_i\xi_j \le \nu^{-1}\sum_{i=1}^n \xi_i^2, \]
where \(\nu>0\). Let
\[ \vec A\in L_{a,a_1}(Q_T),\qquad \vec B\in L_{b,b_1}(Q_T),\qquad C\in L_{c,c_1}(Q_T),\qquad \vec F\in L_{f,f_1}(Q_T), \]
\[ G\in L_{g,g_1}(Q_T), \]
and moreover
\[ \|\vec A\|_{a,a_1,Q_T} + \|\vec B\|_{b,b_1,Q_T} + \|C\|_{c,c_1,Q_T} \le \mu . \]

A generalized solution of equation (8) from the class \(V_2^{1,0}(Q_T)\) \((\dot V_2^{1,0}(Q_T))\) is a function \(u(x,t)\) belonging to \(V_2^{1,0}(Q_T)\) \((\dot V_2^{1,0}(Q_T))\) and satisfying, for all \(t_1,t_2\in[0,T]\) and all \(\Phi(x,t)\in \dot W_2^1(\Omega\times[t_1,t_2])\), the identity
\[ \left.\int_{\Omega} u\Phi\,dx\right|_{t_1}^{t_2} + \int_{t_1}^{t_2}\int_{\Omega} \left[ -u\frac{\partial\Phi}{\partial t} + \left(A_{ij}\frac{\partial u}{\partial x_j}+A_i u\right)\frac{\partial\Phi}{\partial x_i} + \right. \]
\[ \left. + \left(B_i\frac{\partial u}{\partial x_i}+Cu\right)\Phi + F_i\frac{\partial\Phi}{\partial x_j} + G\Phi \right]\,dx\,dt =0. \]

Theorem 1. Let
\[ 1/a+2/na_1=1/n,\quad a_1\in[2,\infty),\quad n\ge2; \]
\[ 1/b+2/nb_1=1/n,\quad b_1\in[2,\infty),\quad n\ge2; \]
\[ 1/c+2/nc_1=2/n,\quad c_1\in[1,\infty),\quad n\ge2; \]
\[ f=2,\quad f_1=2; \]
\[ 1/g+2/ng_1=2/n+\frac12,\quad g_1\in[1,2]\ \text{for } n>2,\quad g_1\in[1,2)\ \text{for } n=2. \]

Then every generalized solution of equation (8) from the class \(\dot V_2^{1,0}(Q_T)\) satisfies the inequality
\[ \|u\|_{V_2^{1,0}(Q_T)} \le c\left[ \|\vec F\|_{f,f_1,Q_T} + \|G\|_{g,g_1,Q_T} + \|u(x,0)\|_{2,\Omega} \right], \]
where
\[ c=c(n,\nu,\mu,a_1,b_1,c_1). \]

Theorem 2. Under the assumptions of Theorem 1 and for any \(\varphi(x)\in L_2(\Omega)\), there exists a unique generalized solution of equation (8) from the class \(\overset{\circ}{V}{}^{1,0}_2(Q)\), equal to \(\varphi(x)\) for \(t=0\). Every generalized solution from this class satisfies the condition
\[ \|u(x,t+h)-u(x,t)\|_{2,2,Q_T}=o(h^{1/2}). \]

Denote by \(\Gamma\) the set of points lying on the lateral surface and the lower base of the cylinder \(Q_T\). Let \(Q'\) denote an arbitrary domain contained in \(Q_T\) and at positive distance from \(\Gamma\).

Theorem 3. Let
\[ 1/a+2/na_1=1/n,\quad a_1\in[2,\infty),\quad n\geqslant2; \]
\[ 1/b+2/nb_1=1/n,\quad b_1\in[2,\infty),\quad n\geqslant2; \]
\[ 1/c+2/nc_1=2/n,\quad c_1\in[1,\infty),\quad n\geqslant2; \]
\[ 1/f+2/nf_1=1/n+\sigma/2,\quad \sigma\in(0,1),\quad f_1\in[2,2/\sigma]\ \text{for } n>2,\quad f_1\in[2,2/\sigma)\ \text{for } n=2; \]
\[ 1/g+2/ng_1=2/n+\theta/2,\quad \theta\in(0,1),\quad g_1\in[1,2/\theta]\ \text{for } n>2,\quad g_1\in[1,2/\theta)\ \text{for } n=2. \]

Then an arbitrary generalized solution of equation (8) from the class \(V^{1,0}_2(Q_T)\) belongs to the space \(L_{\chi,\chi_1}(Q')\), where
\[ 1/\chi+2/n\chi_1=\lambda/2,\quad \chi_1\in[2/\lambda,\infty]\ \text{for } n>2,\quad \chi_1\in(2/\lambda,\infty]\ \text{for } n=2,\quad \lambda=\max(\sigma,\theta). \]
If this generalized solution is bounded on \(\Gamma\), then it also belongs to \(L_{\chi,\chi_1}(Q_T)\) with the same exponents \(\chi\) and \(\chi_1\).

Theorem 4. Suppose that, in the assumptions of Theorem 3, the exponents \(\sigma\) and \(\theta\) are equal to zero.

Then an arbitrary generalized solution of equation (8) from the class \(V^{1,0}_2(Q_T)\) belongs to the space \(L_{\chi,\chi_1}(Q')\) with arbitrary \(\chi\) and \(\chi_1\). If this generalized solution is bounded on \(\Gamma\), then it also belongs to \(L_{\chi,\chi_1}(Q_T)\).

Theorem 5. Let
\[ 1/a+2/na_1=1/n-\varepsilon,\quad \varepsilon>0,\quad a_1\in(2,\infty),\quad n\geqslant2; \]
\[ 1/b+2/nb_1=1/n,\quad b_1\in[2,\infty),\quad n\geqslant2; \]
\[ 1/c+2/nc_1=2/n-\delta,\quad \delta>0,\quad c_1\in(1,\infty),\quad n\geqslant2; \]
\[ 1/f+2/nf_1=1/n,\quad f_1\in[2,\infty]\ \text{for } n>2,\quad f_1\in[2,\infty)\ \text{for } n=2; \]
\[ 1/g+2/ng_1=2/n,\quad g_1\in[1,\infty]\ \text{for } n>2,\quad g_1\in[1,\infty)\ \text{for } n=2. \]

Then there exists a constant \(d\), depending only on \(n,\nu,\mu\), such that the integrals
\[ E_{Q'}(u)=\iint_{Q'} \exp\{d(\|u\|_{2,2,Q_T}+1)^{-1}|u(x,t)|\}\,dx\,dt \]
are finite for any generalized solution of equation (8) from the class \(V^{1,0}_2(Q_T)\). If this solution is bounded on \(\Gamma\), then the integral \(E_{Q_T}(u)\) is also finite.

Theorem 6. Let
\[ 1/a+2/na_1=1/n-\varepsilon,\quad \varepsilon>0,\quad a_1\in(2,\infty),\quad n\geqslant2; \]
\[ 1/b+2/nb_1=1/n,\quad b_1\in[2,\infty),\quad n\geqslant2; \]
\[ 1/c+2/nc_1=2/n+\delta,\quad \delta>0,\quad c_1\in(1,\infty),\quad n\geqslant2; \]
\[ 1/f+2/nf_1=1/n-\sigma,\quad \sigma>0,\quad f_1\in(2,\infty]\ \text{for } n>2,\quad f_1\in(2,\infty)\ \text{for } n=2; \]
\[ 1/g+2/ng_1=2/n-\theta,\quad \theta>0,\quad g_1\in(1,\infty]\ \text{for } n>2,\quad g_1\in(1,\infty)\ \text{for } n=2. \]

Then every generalized solution of equation (8) from the class \(V^{1,0}_2(Q_T)\) is bounded in every domain \(Q'\). If this generalized solution is bounded on \(\Gamma\), then it will also be bounded in the whole cylinder \(Q_T\).

Theorem 7. Under the assumptions of Theorem 6, an arbitrary generalized solution of equation (8) from the class \(V^{1,0}_2(Q_T)\) is Hölder continuous in any domain \(Q'\). If, in addition, such a generalized solution is Hölder continuous on \(\Gamma\), then it will have a bounded Hölder constant in the whole cylinder \(Q_T\).

We now consider the quasilinear equation
\[ \frac{\partial u}{\partial t}-\frac{d}{dx_i}\bigl(A_i(x,t,u,u_x)\bigr)+A(x,t,u,u_x)=0, \tag{9} \]
in which the functions \(A_i(x,t,u,\vec p)\) and \(A(x,t,u,\vec p)\) are defined for \((x,t)\in Q_T\), \(|u|\leqslant M\), and arbitrary \(p=(p_1,\ldots,p_n)\), are continuous in \(u\) and \(p_k\), and satisfy the conditions
\[ A_i p_i\geqslant \nu|\vec p|^2-\varphi^{(1)},\qquad |A_i|\leqslant \mu|\vec p|+\varphi^{(2)},\qquad |A|\leqslant \mu|\vec p|^2+\varphi^{(3)}, \tag{10} \]
where \(\nu,\mu>0\), \(\varphi^{(1)}\in L_{q,q_1}(Q_T)\), \(\varphi^{(2)}\in L_{r,r_1}(Q_T)\), \(\varphi^{(3)}\in L_{s,s_1}(Q_T)\), \(\varphi^{(i)}\geqslant0\), \(i=1,2,3\).

Theorem 8. Let \(1/q+2/nq_1=2/n-\varepsilon\), \(\varepsilon>0\), \(q_1\in(1,\infty]\) for \(n>2\), \(q_1\in(1,\infty)\) for \(n=2\); \(1/r+2/nr_1\leq 1/n-\delta\), \(\delta>0\), \(r_1\in(2,\infty]\) for \(n>2\), \(r_1\in(2,\infty)\) for \(n=2\); \(1/s+2/ns_1=2/n-\chi\), \(\chi>0\), \(s_1\in(1,\infty]\), \(n>2\), \(s_1\in(1,\infty)\), \(n=2\).

Then any bounded generalized solution from the class \(V_2^{1,0}(Q_T)\) of equation (9)\(^*\), satisfying conditions (10), belongs to the Hölder space \(C_{x,t}^{\alpha,\alpha/2}(Q')\) with some \(\alpha>0\) for any domain \(Q'\subset \overline{Q'}\subset Q_T\).

Finally, let \(u(x,t)\) be a solution of the equation

\[ \frac{\partial u}{\partial t} - A_{ij}(x,t,u,u_x)u_{x_i x_j} + A(x,t,u,u_x) = 0, \]

in which the functions \(A_{ij}(x,t,u,\vec p)\), \(A(x,t,u,\vec p)\) are defined in a neighborhood of the manifold
\[ \mathfrak M=\{(x,t)\in \overline{Q_T},\ u=u(x,t),\ \vec p=u_x(x,t)\} \]
and are continuous in \(u\) and \(p\). Suppose further that \(A_{ij}(x,t,u,p)\) are continuously differentiable with respect to \(x,u\), and \(p_k\), and that on the solution \(u(x,t)\) the functions \(A_{ij}\) and \(A\) satisfy the conditions

\[ \nu\sum_{i=1}^{n}\xi_i^2 \leq A_{ij}(x,t)\xi_i\xi_j \leq \nu^{-1}\sum_{i=1}^{n}\xi_i^2, \qquad \nu>0, \]

\[ \left|\partial A_{ij}/\partial p_k\right|\leq \mu, \qquad \left|\partial A_{ij}/\partial u;\ \partial A_{ij}/\partial x_k,\ A\right| \leq \varphi(x,t), \qquad \varphi(x,t)\geq 0, \]

\[ \varphi\in L_{q,q_1}(Q_T),\qquad 1/q+2/nq_1=1/n-\varepsilon,\qquad \varepsilon>0,\qquad q_1\in(2,\infty),\quad n\geq 2. \]

Theorem 9. Under the assumptions made above, the Hölder norm
\[ |u_x|_{C_{x,t}^{\alpha,\alpha/2}(Q')} \]
with some \(\alpha>0\) is estimated solely in terms of \(n,\nu,\mu,q,q_1\), the distance from \(Q'\) to \(\Gamma\), and
\[ \max_{Q_T}|u_x|. \]

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
25 VI 1965

REFERENCES

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\(^*\) That is, any function \(u(x,t)\) belonging to \(V_2^{1,0}(Q_T)\), satisfying for all \(t_1,t_2\in[0,T]\) and bounded \(\Phi(x,t)\in W_2^1(\Omega\times[0,T])\) the integral identity

\[ \left.\int_{\Omega}u\Phi\,dx\right|_{t_1}^{t_2} + \int_{t_1}^{t_2}\int_{\Omega} \left[A_i\Phi_{x_i}+A\Phi\right]\,dx\,dt = 0 \]
and such that \(|u(x,t)|\leq M\).