\documentclass[12pt]{article} \usepackage{amsmath,amssymb,latexsym} \usepackage[cp1251]{inputenc} \usepackage{amsmath,amsthm,lscape,floatflt} \usepackage[english]{babel} \vfuzz2pt \hfuzz2pt \theoremstyle{plain} \newtheorem*{theorem}{Theorem} \newtheorem*{corollary}{Ñorollary} \newtheorem{lemma}{Lemma} \newtheorem{proposition}{Proposition} \theoremstyle{definition} \newtheorem{definition}{Definition} \theoremstyle{remark} \newtheorem*{notation}{Notation} \theoremstyle{remark} \newtheorem{remark}{\textbf{Remark}} \def\oint{\operatorname{\int\!\!\!\!\!--}} \def\dint{\operatorname{\int\!\!\!\!\!=}} %\pagestyle{empty} \begin{document} \centerline{\large\bf New a priori estimates for nondiagonal} \centerline{\large\bf strongly nonlinear parabolic systems} \medskip \centerline{\bf Arkhipova A.} \medskip \centerline{\it Saint-Petersburg State University,Russia} \centerline{\it Arina@AA1101.spb.edu} \bigskip \centerline {\bf Introduction} \medskip We consider the Cauchy-Dirichlet problem for nonlinear parabolic systems. Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $n\geq2$, with sufficiently smooth boundary $\partial\Omega$, let $Q=\Omega\times(0,T)$ with any fixed $T>0$, and $u:Q\mapsto\mathbb{R}^N,\,\,u=(u^1,...,u^N),\, N>1,\,$ be a solution of the problem \begin{align} u^k_t-(A^{\alpha\beta}_{kl}(z,u)u^l_{x_\beta})_{x_\alpha}+b^k(z,u,u_x)=0,\quad k=1,...,N,\quad z\in Q, \label{E:1} \end{align} \begin{align} u|_{\Gamma}=0,\quad\, u|_{t=0}=\phi(x), \label{E:2} \end{align} where $u_x=\{u^k_{x_\alpha}\},\quad \Gamma=\partial\Omega\times(0,T).$ It is assumed that the matrix $A=\{A^{\alpha \beta}_{kl}(\cdot,\cdot)\}^{\alpha,\beta \leq n}_{k,l\leq N}$ is defined on the set $\mathcal{M}=\overline{Q}\times\mathbb{R}^N\,$, the function $b=\{b^k(\cdot,\cdot,\cdot)\}^{k\leq N}\,$ is the Caratheodory function on $\mathcal{M}\times\mathbb{R}^{nN}$. Moreover, we suppose the following: \medskip \noindent \textbf{a)} the strong parabolicity condition holds, i.e., there exist numbers $\nu$ and $ \mu >0$ such that for every $(z,u)\in\mathcal{M},\quad \xi\in\mathbb{R}^{nN}$, \begin{align} (A(z,u)\xi,\xi)\geq\nu |\xi|^2, \label{E:3} \end{align} \begin{align} \underset{\mathcal{M}}{\sup}\|A(z,u)\|\leq\mu, \label{E:4} \end{align} \medskip \textbf{b)} $A^{\alpha\beta}_{kl} $ are uniformly continuous functions, more exactly, there is a function $\omega(s,\tau)$ defined and continuous on $[0,\infty)\times[0,\infty)\,\,$ which is bounded, nondecreasing, concave in $\tau$ for any fixed $s$, $\omega(0,0)=0$, and such that \begin{align} \| A(z,u)-A(\zeta,v)\| \leq \omega(|z-\zeta|^2,|u-v|^2)\quad z,\zeta\in\overline{Q}, \quad u,v\in\mathbb{R}^{N}, \label{E:5} \end{align} \noindent \textbf{c)} function $b$ satisfies the growth condition \begin{align} |b(z,u,p)|\leq b_0|p|^2+\mu,\quad (z,u)\in\mathcal{M},\,\,p\in\mathbb{R}^{nN}, \, b_0=const>0, \label{E:6} \end{align} \textbf{d)} $\Omega $ is a strongly Lipschitz domain, $\phi\in\overset{0}{W^1_2}(\Omega)$. \medskip Condition (\ref{E:6}) defines strongly nonlinear term $b(z,u,u_x)$ in system ( \ref{E:1}). It is known that in the case of smooth data and under compatibility conditions, there exists a classical solution of (\ref{E:1}),(\ref{E:2}) on some time interval $[0,T_0)$ (see, for example, Theorem 1, ~\cite{AA1}). A singular set $\sigma\subset\overline{\Omega}$ for $t=T_0$ is estimated in ~\cite{AA2}. It is proved in ~\cite{AA2} that (n-2)-Hausdorff measure of $\sigma$ is finite $(\mathcal{H}_{n-2}(\sigma)<+\infty)$. Strong nonlinearity (\ref{E:6}) does not allow to apply well-known abstract theorems to state weak global solvability of the problem under consideration. Global solvability of this problem was proved only for a particular case. One class of systems was studied by the author in the case of two spatial variables ( ~\cite{AA1},~\cite{AA3}-~\cite{AA5}). It is assumed in these papers that the elliptic operator of a system is of a variational structure, and conditions \textbf{a)}-\textbf{c)} are valid. Existence of almost everywhere smooth in $\overline{Q}$ solution was proved under the Dirichlet and Neumann boundary conditions. The solution may have at most finitely many singular points. It was a development of the main idea by M.Struwe ~\cite{Str7}. In the situation $b_0=0$ in (\ref{E:6}), weak solvability of (\ref{E:1}),(\ref{E:2}) is a consequence of the Monotone Operators theory. In this case, it is interesting to study regularity problem. Counterexamples show that in the multidimensional case, one can expect only partial regularity of solutions even under all smooth data of the problem (~\cite{St-J-M9},~\cite{J-St8}). As about the two dimensional case, it is unknown up to now whether a solution of the simplest system \begin{align*} u^k_t-(A^{\alpha\beta}_{kl}(u)u^l_{x_\beta})_{x_\alpha}=0,\quad,k\leq N,\quad z\in Q, \end{align*} is smooth in $\overline{Q}$ for any $T>0$ ($ A^{\alpha\beta}_{kl},\,u|_{\partial'Q}$ are smooth enough) or a singular set can appear for $t>0$. On the other hand, when $b_0\neq 0$ and the principal matrix $A$ is of the diagonal structure, i.e., \begin{align} A^{\alpha\beta}_{kl}(z,u)=a^{\alpha\beta}(z,u)\delta_{k,l}, \label{E:7} \end{align} where $\delta_{k,l}$ is the Croneker symbol, global solvability of (\ref{E:1}),(\ref{E:2}) was proved only for special cases. Structural restrictions on function $b(z,u,u_x)\,$ were formulated in the monograph ~\cite{L-S-U10}, (Chapter 7, $\S7$) to prove global classic solvability or existence in the class $L^2((0,T);\overset{0}{W^1_2}(\Omega))\bigcap\mathcal{C}^{\alpha, \frac{1}{2}\alpha}(\overline{Q}),\,\,\alpha\in(0,1),$ (see also ~\cite{T11},~\cite{W12} on more sharp conditions for $b$). The systems of the type(\ref{E:1}),(\ref{E:7}), $a^{\alpha\beta}=a^{\alpha\beta}(x)$, describe heat flows of harmonic maps. Existence of global almost everywhere smooth in $\overline{Q}$ solution (it has finite energy and satisfies the integral identity in the sense of distributions) was proved for the two dimensional case ~\cite{Str7}, and for the multidimensional case ~\cite{Ch-Str6}. It was stated that the solution may have at most finitely many singular points in the case $n=2$, and $\mathcal{H}_n(\Sigma;\delta)<+ \infty$ for $n>2$. We also mention the corresponding stationary case of the problem (\ref{E:1}),(\ref{E:2}). For more general class of the elliptic operators \begin{align*} L=\{L^k\}^{k\leq N}, \quad L^k=-(a^k_\alpha(x,u,u_x))_{x_\alpha}+b^k(x,u,u_x), \end{align*} ($b\,$ satisfies condition (\ref{E:6}), $n=\,2$), the Dirichlet problem was studied by J. Frehze ~\cite{Fr13}. Under the so-called "one-side condition" \begin{align*} b(x,u,p)\cdot u\geq \nu_ {\ast}|p|^2-\mu;\quad \nu_{\ast}<\nu, \end{align*} existence of a solution $u\in\overset{0}{W^1_2}(\Omega)\cap\mathcal{C}^{\alpha}(\overline{\Omega}),\,\alpha\in(0,1)$ was proved in ~\cite{Fr13}. \textit{Regularity} problem for elliptic and parabolic nondiagonal systems of equations with strongly nonlinear terms in the gradient was studied in ~\cite{G-G26},~\cite{G-Str14},~\cite{M-M15}, ~\cite{M-M16}. Partial regularity of \textit{bounded} weak solutions was proved. In the parabolic situation, under the assumption \begin{equation} 2b_0\|u\|_{\infty,Q}<\nu, \label{E:8} \end{equation} it was proved that the solution $u$ of system (\ref{E:1}) is the H\"{o}lder continuous function in the vicinity of a point $z^0\in Q\,$ provided that \begin{align} \liminf_{R\rightarrow 0}\frac{1}{R^n}\underset{Q_R(z^0)}{\int}|u_x|^2\,dz<\varepsilon^2_0, \label{E:9} \end{align} where number $\varepsilon_0>0$ depends on the data only. It means that condition (\ref{E:9}) describes regular points of bounded weak solutions of (\ref{E:1}) under restrictions (\ref{E:8}) inside $Q$. Regularity of bounded weak solutions near the lateral surface $\Gamma,$ $\Gamma=\partial\Omega\times(0,T),\,$ under the Dirichlet and Neumann boundary conditions and restriction (\ref{E:8}) was studied in ~\cite{A17}. It was proved partial regularity of the solutions up to $\Gamma$, and was estimated the Hausdorff measure of the corresponding singular sets. Note that to prove smoothness of solution $u$ in a neighborhood of $z^0$, it is sufficient to assume that \begin{align} \underset{Q_{R_0}(z^0)\cap\overline{Q}}{osc}u<\theta \label{E:10} \end{align} for some $\theta>0$ and $R_0=R_0(z^0)>0$ (instead of conditions (\ref{E:8}),(\ref{E:9})). Indeed, from the integral identity for problem (\ref{E:1}),(\ref{E:2}), and assumption (\ref{E:10}), it follows that \begin{align} \frac{1}{R^n}\underset{Q_R(z^0)}{\int}|u_x|^2\,dz\leq c_0(\theta^2+R^2), \label{E:11} \end{align} where the constant $c_0$ does not depend on $z^0,\,R\leq\frac{R_0}{2}$. Evidently, that the inequalities \begin{equation*} \theta<\frac{\nu}{b_0},\quad c_0(\theta^2+R^2_0)<\varepsilon^2_0, \end{equation*} guarantee condition (\ref{E:9}) and local variant of (\ref{E:8}) ($ b_0\underset{Q_{R_0}}{osc}\,u <\nu $). Therefore, all mentioned results on the regularity are valid under condition (\ref{E:10}). Moreover, this condition allows us to estimate the stronger norms of $u$ in the vicinity of $z^0$. Unfortunately, description (\ref{E:10}) of regular points does not allow to obtain reasonable estimate of the set of singular points of the solution under consideration. At the same time, an appropriate information on the singular set can be helpful to study \textit{solvability} of the problem (see, for example, ~\cite{Ch-Str6}). There arises a question how to relax condition (\ref{E:10}) in description of regular points. The author studied this problem for the stationary case. It was considered the Dirichlet problem for quasilinear elliptic systems with quadratic nonlinearities in the gradient. We proved in ~\cite{A18} and ~\cite{A19} that the assumption \begin{align} \underset{\Omega_R(x^0)}{osc}u < \theta,\quad \Omega_R(x^0)=\Omega\cap B_R(x^0), \label{E:12} \end{align} in a point $x^0\in\overline{\Omega}\,$, supplying an estimate of the H\'older norm of $u\,$, can be relaxed to condition \begin{align} [u]_{\mathcal{L}^{2,n}(\Omega_R(x^0))}+ \underset{y^0\in\partial\Omega\cap B_R(x^0),\, \rho\leq R}{sup}|u_{\rho,y^0}|<\theta. \label{E:13} \end{align} We denote by $[\cdot]_{\mathcal{L}^{2,n}(\Omega_R)}$ the seminorm in the Campanato space $\mathcal{L}^{2,n}(\Omega_R)$. The second term in (\ref{E:13}) is absent for points $x^0$ inside $\Omega$. Instead of (\ref{E:13}), we can assume the condition \begin{align} \|u_x\|_{L^{2,n-2}(\Omega_R(x^0))}<\theta_1. \label{E:14} \end{align} Numbers $\theta$ and $\theta_1$ in (\ref{E:13}) and (\ref{E:14}) depend on the data of the problem, $\theta_1= c(n)\theta$. In the situations, when (\ref{E:14}) can be relaxed to the condition: \begin{align} \frac{1}{R^{n-2}}\underset{\Omega(x^0)}{\int} |u_x|^2\,dx<\theta_1 \label{E:15} \end{align} in the fixed point $x^0\in\overline{\Omega}$ for sufficiently small $R$, we are able to estimate the Hausdorff measure of the singular set. In the case of the simplest quasilinear elliptic systems ($b_0=0\,$) , condition (\ref{E:15}) describes regular points of solutions. That is why we can say that (\ref{E:15}) introduces an optimal description of regular points in the case $b_0\neq 0$. There arises a question of the possibility to relax condition (\ref{E:14}) to conditions (\ref{E:15}). It is not difficult to see that the monotonicity of the function \begin{align} \Phi(\rho,y)=\frac{1}{\rho^{n-2}}\underset{\Omega_\rho(y)}{\int}|u_x|^2\,dx \label{E:16} \end{align} in $\rho$ for any fixed $y\in\overline{\Omega}\,$ provides such transformation. For one class of strongly nonlinear elliptic systems, the author proved monotonicity type inequality for function $ \Phi$ introduced by (\ref{E:16}) ~\cite{A20}. This inequality permitted to relax condition (\ref{E:14}) to the optimal regularity condition (\ref{E:15}). It is evidently that in the stationary situation, suggested by the author condition (\ref{E:14}) allowed us to transform the problem of the optimal description of the regular set to derivation of the monotonicity type inequality for function $\Phi(\rho,\cdot)$. The same considerations have been undertaken by the author for the parabolic problem (\ref{E:1}),(\ref{E:2}). The question was how to relax condition (\ref{E:10}) to describe regular points of solutions. For a solution $u$ of (\ref{E:1}),(\ref{E:2}), a local $L_p$-estimate, $p>2$, of the gradient was obtained under condition \begin{align} [u]_{\mathcal{L}^{2,n+2}(Q_R(z^0))}+\underset{\zeta\in \Gamma\cap Q_R,\,\rho\leq R}{sup}|u_{\rho,\zeta}|<\theta, \label{E:17} \end{align} or \begin{equation} \|u_x\|_{L^{2,n}(Q_R(z^0))}<\theta_1,\quad z^0\in Q\cap\Gamma, \label{E:18} \end{equation} $\theta_1=c(n)\theta\,$ (see ~\cite{A21}, Theorem 2.1 and Remark 2.3). We note that the $L_p$ - estimate of $u_x$ is useful to estimate the H\"{o}lder norm of $u$ locally in the frame of the so-called direct method. We explain in this paper how to estimate $u$ in $\mathcal{C}^{\alpha, \frac{\alpha}{2}}$- norm, $\alpha\in (0,1)$, in a neighborhood of a fixed point $z^0\in Q\cup\partial'Q\,$ provided that condition (\ref{E:17}) (or (\ref{E:18})) holds. Finally, we would like to remark that some smoothness of the solution under consideration is assumed in this work. We intend to apply all a priori information about the solution to investigate in the future some regularization of the problem. \bigskip We adopt the following notation: \begin{align*} &\Lambda=(0,T),\quad Q=\Omega\times(0,T),\,\, B_R(x^0)=\{x\in\mathbb{R}^n: \,|x-x^0|^{(\alpha,\frac{\alpha}{2})}_Q = \underset{z,z'\in\overline{Q},\,z\neq z'}{\sup}\frac{|u(z)-u(z')|}{\delta(z,z')^\alpha}.$ \medskip \noindent Let the Lipschitz characteristics of $ \partial\Omega$ are estimated by the constant $l_\Gamma.$ \bigskip \centerline{\bf A priori estimate of the H\"{o}lder norm} \medskip \textbf{1}. \textit{$L_p$-estimate of the gradient of a solution.} \medskip Let conditions \textbf{a), c),} and \textbf{d)} of $\S$ 1 hold. By Theorem 2.1 and Remark 2.3 ~\cite{A21}, there exists a number $\theta>0$ such that condition (\ref{E:17}) (or (\ref{E:18})) for a fixed point $z^0\in Q\cap\Gamma$ ensures the estimate \begin{equation} \bigl(\underset{Q_\rho(\xi)}{\oint}|u_x|^p\,dz\bigr)^{\frac{2}{p}}\leq c_1\underset{Q_{a\rho}(\xi)}{\oint} (1+|u_x|)^2\,dz \label{E:19} \end{equation} for all $\xi\in Q_{\frac{R}{2}}(z^0),\quad \rho\leq \frac{R}{2a},\,\,a\geq 2 $ is an absolute constant. Here $p=p(\nu, \mu,n)>2,\quad c_1=c_1(\nu,\mu,n,\theta,l_\Gamma)$. \begin{remark} To derive (\ref{E:19}), it was assumed in ~\cite{A21} that $u\in L_\infty(\Lambda;L_2(\Omega))\cap L_2(\Lambda;\overset{0}{W^1_2}(\Omega))$, and there exists $m>2\,$ such that $u_x\in L_m(Q_R(z^0))$. Moreover, it follows from (\ref{E:17}) or (\ref{E:18}) that the corresponding characteristics of $u$ should be finite. Certainly, $p\leq m$ in (\ref{E:19}). Further, we say that \textit{" $u$ is a suitable solution in $Q_R(z^0)$" } provided that all mentioned in Remark 1 characteristics of $u\,$ are finite in $Q_R(z^0)$. \end{remark} \begin{remark} In (\ref{E:19}) and below we denote by $c_i$ different constants that may depend on the parameters of the data. Dependence on other parameters we mark explicitly. For example, $c_i=c_i(\varepsilon), \varepsilon > 0$. \end{remark} We did not discuss in ~\cite{A21} the situation $\mathbb{P}_R(z^0)\cap\{t=0\}\neq \emptyset.$ The corresponding analysis was done earlier in ~\cite{A22}. We studied in ~\cite{A22} bounded weak solutions $u$ of (\ref{E:1}),(\ref{E:2}), $u\in V=\{v:\underset{\Lambda}{esssup}\, \|v(\cdot,t)\|_{2,\Omega}+\|v_x\|_{2,Q}< +\infty\}$ and assumed condition (\ref{E:8}). The reverse H\"{o}lder inequalities for $|u_x(z)|$ with additional terms depending on the initial function $\phi$ were derived in ~\cite{A22}. Using the same idea, we are able to deduce in our situation the quasireverse H\"{o}lder inequalities for $g(z)=(1+|u_x|)^{\frac{2}{q}},\quad q=\frac{n+2}{n}$: \begin{align} \underset{Q_\rho(\xi)}{\oint}g^q\,dz\leq \varepsilon_1\underset{Q_{a\rho}(\xi)}{\oint}g^q\,dz + c_2(\varepsilon_1)\bigl(\underset{Q_{a\rho}(\xi)}{\oint}g\,dz\bigr)^q \notag \\ + c_3(l)\theta b_0 \bigl(\underset{Q_{a\rho}(\xi)}{\oint}g^{ql}\,dz\bigr)^{\frac {1}{l}} + c_4\bigl(\underset{\Omega^{(0)}_{a\rho}(x^\ast)}{\oint}|\phi_x|^{\frac{2}{q}}\,dx\bigr)^q,\notag\\ \xi=(x^\ast,t^\ast),\, \xi\in \overline{\Omega_{\frac{R}{2}}(x^0)}\times\bigl[0,t^0+\bigl(\frac{R}{2}\bigr)^2\bigr],\,\rho\leq\frac{R}{2a},\, a=const\geq 2. \label{E:20} \end{align} We follow ~\cite{A22}, ~\cite{A29}, and ~\cite{A21} to assert that \begin{align*} \bigl(\underset{Q_\rho(\xi)}{\oint}|u_x|^p\,dz\bigr)^{\frac{2}{p}}\leq c_5\underset{Q_{a\rho}(\xi)}{\oint}((1+|u_x|)^2\,dz \notag\\+ c_6\bigl(\underset{\Omega^{(0)}_{a\rho}(x^\ast)}{\oint}|\phi_x|^{\frac{2}{q}+(p-2)}\,dx\bigr)^{\frac{2}{\frac{2}{q}+(p-2)}}, \end{align*} for some $p=p(\nu,\mu,n)>2$ and the same $\xi$ and $\rho$ as in (\ref{E:20}). We may suppose that $p < \frac{2(n+4)}{n+2}$, apply the H\"{o}lder inequality to the last inequality, and derive the following estimate: \begin{align} \bigl(\underset{Q_\rho(\xi)}{\oint}|u_x|^p\,dz\bigr)^{\frac{2}{p}}\leq c_5 \underset{Q_{a\rho}(\xi)}{\oint}(1+|u_x|)^2\,dz + c_6 \underset{\Omega^{(0)}_{a\rho}(x^\ast)}{\oint}|\phi_x|^2\,dx,\notag\\ \rho\leq\frac{R}{2a}. \label{E:21} \end{align} In (\ref{E:21}) $\xi=(x^\ast,t^\ast)\in \overline{\Omega_{\frac{R}{2}}(x^0)}\times [0,t^0+ (\frac{R}{2})^2]\,$ for the case $t^0\leq R^2,$ and $\xi\in \overline{Q_{\frac{R}{2}}(z^0)}$ in the other case. \medskip \textbf{2}. \textit{On the energy estimates for solutions (\ref{E:1}),(\ref{E:2})}. \medskip First, we remark that the global energy estimate for a \emph{bounded} solution $ u \in L_\infty(\Lambda;L_2(\Omega))\cap L_2(\Lambda;\overset{0}{W^1_2}(\Omega))$ one can deduce provided that condition (\ref{E:8}) holds. Instead of (\ref{E:8}), we can assume "one-side condition" mentioned earlier, and derive both global and local variant of the energy estimate: \begin{align*} \underset{\Lambda_R(t^0)}{\sup} \|u(\cdot,t)\|^2_{2,\Omega(x^0)} + (\nu-\nu_\ast)\|u_x\|^2_{2,Q_R(z^0)}\leq \mu |Q_{2R}| \\ + \frac{c}{R^2}\underset{Q_{2R}(z^0)}{\int}|u_x|^2\,dz + \|\phi\|^2_{2,\Omega^{(0)}_{2R}(x^0)},\quad z^0\in \overline{Q}. \end{align*} Nevertheless, we are forced to assume validity of the stronger energy estimates for our future considerations. In ~\cite{A17}, local estimate of the H\"{o}lder norm of solution $u$ was derived under condition (\ref{E:8}) and assumption that $u_t\in L_2(Q)$. In general, the last condition is very strong but for the parabolic systems with the elliptic operators of the variational structure, the following estimate holds: \begin{align} \|u_t\|^2_{2,Q} +\underset{\Lambda}{\sup}\|u_x(\cdot,t)\|^2_ {2,\Omega}\leq c_7(1+\|\phi\|^2_{2,\Omega}). \label{E:22} \end{align} More exactly, let $L$ be the Euler operator for the quadratic functional \begin{align} E[u]=\underset{\Omega}{\int}\bigl[\frac1{2}A^{\alpha\beta}_{kl}(x,u)u^l_{x_\beta}u^k_{x_\alpha} + f^k(x)u^k\bigr]\,dx. \label{E:23} \end{align} Then \begin{align*} b^k(x,u,u_x)= \frac{1}{2} (A^{\alpha\beta}_{ml}(x,u))'_{u^k}u^l_{x_\beta}u^m_{x_\alpha} + f^k(x) \end{align*} in system (\ref{E:1}), and \begin{align*} b_0=\frac {1}{2}\underset{x\in \Omega,\,u\in \mathbb{R}^N}{\sum}\,\underset{\alpha,\beta\leq n;\,m,k,l\leq N}{\sum}\bigl|(A^{\alpha\beta}_{ml}(x,u))'_{u^k}\bigr|<\infty,\notag\\ \mu=\|f\|_{\infty,\Omega}\quad for \quad f\in L_{\infty}(\Omega). \end{align*} For the system $u_t + Lu = 0\,$ with the described operator $L$ (under conditions ( \ref{E:2})), inequality (\ref{E:22}) can be obtained immediately. Moreover, local energy estimate \begin{align} \underset{\Lambda_r(\hat{t})}{\sup}(\underset{\Omega_r(\hat{x})}{\int}|u_x(x,t)|^2\,dx) + \underset{Q_r(\hat{z})}{\int}|u_t|^2\,dz \leq \notag\\ c_8\bigl\{\frac{1}{r^2}\underset{Q_{2r}(\hat{z})}{\int}(1+|u_x|)^2\,dz + \underset{\Omega^{(0)}_{2r}(\hat{x})}{\int}|\phi_x|^2\,dx\bigr\},\quad \hat{z}=(\hat{x},\hat{t})\in \overline{Q},\quad r\leq d_0, \label{E:24} \end{align} also holds. In particular, it follows from (\ref{E:24}) that \begin{align} \underset{Q_r(\hat{z})}{\oint}|u-u_{r,\hat{z}}|^2\,dz \leq c_9\bigl(\underset{Q_{2r}(\hat{z})}{\dint}(1+|u_x|)^2\,dz \notag\\+ \underset{\Omega^{(0)}_{2r}(\hat{x})}{\dint}|\phi_x|^2\,dx \bigr). \label{E:25} \end{align} Further we assume that inequality (\ref{E:24}) holds for the solution under investigation. Now we are ready to formulate the main result of the paper. \medskip \textbf{3}. \textit{ The main result}. \medskip \begin{theorem} Let conditions \textbf{a)-d)} hold, let $u\,$ be a suitable solution of (\ref{E:1}), (\ref{E:2}) in $Q_{R_0}(z^0)\subset Q $ (see Remark 1), and satisfy inequality (\ref{E:24}) in all cylinders of $Q_{R_0}(z^0)$. Let $\gamma_{R_0}(x^0)=B_{R_0}(x^0)\cap\partial\Omega\in\mathcal{C}^1\,$ and $\phi_x\in L^{2,n-2+2\alpha}(\Omega^{(0)}_{R_0}(x^0))\,$ for a fixed $\alpha\in (0,1)$. There exist positive numbers $\theta$ and $R\leq R_0$ such that the assumption \begin{align} \|u_x\|_{L^{2,n}(Q_R(z^0);\delta)} < \theta \label{E:26} \end{align} guaranties the estimate \begin{align} \langle u\rangle^{(\beta,\frac{\beta}{2})}_{Q_{\frac{R}{2}}(z^0)}\leq c_{10}(1+ \|u_x\|_{2,Q_R(z^0)}). \label{E:27} \end{align} Here $\beta$ is an arbitrary number in $(0,1)$ provided that $\Omega^{(0)}_R(x^0) = \emptyset$, and $\beta\leq \alpha$ in the case $\Omega^{(0)}_R(x^0)\neq\emptyset$. The constant $c_{10}$ depends on $R^{-1}, \nu, \mu, b_0, \theta, \beta, n$. Moreover, $c_{10}$ may also depend on $\|\phi_x\|_{L^{2,n-2+2\alpha}(\Omega^{(0)}_R(x^0))}$ and $\mathcal{C}^1$- characteristics of $\gamma_R(x^0)$. \end{theorem} \medskip \textbf{4}. \textit{Model setting of the problem}. \medskip The most interesting situations of the disposition of $z^0\in Q\cap\partial'Q$ are the following: $\Gamma_R(z^0)=\mathbb{P}_R(z^0)\cap\Gamma\neq\emptyset,\quad$ and $\Omega^{(0)}_R(x^0)=\mathbb{P}_R(z^0)\cap\{t=0\}\neq\emptyset$ ($z^0$ is close to the parabolic boundary of $Q$). We consider this case below. Let $y^0$ be the nearest point to $x^0$ at $\partial\Omega$. We introduce $\mathcal{C}^1$ - diffeomorphism $y=y(x)$ of some neighborhood $V(y^0)$ so that $V(y^0)\cap\partial\Omega\subset\gamma_{R_0}(x^0),\,\,x^0\in V(y^0)\cap\overline{\Omega}\,$ and $y(V(y^0)\cap\Omega)=B^+_1(0),\,\,$ $y(V(y^0)\cap\partial\Omega)=\gamma_1(0).$ Here and below $\gamma_r(0)=B_r(0)\cap\{y_n=0\}$. The function $\tilde{u}(y,t)=u(x(y),t)$ is a solution of the problem \begin{align} \tilde{u}^k_t -\bigl(\mathcal{A}^{\alpha\beta}_{kl}(\xi,\tilde{u})\tilde{u}^l_{y_\beta}\bigr)_{y_\alpha} + D^k(\xi,\tilde{u},\tilde{u}_y) = 0,\notag\\ \tilde{u}|_{\Gamma_1} = 0,\quad \Gamma_1=\gamma_1(0)\times(0,T),\quad \xi=(y,t)\in B^+_1(0)\times(0,T),\notag\\ \tilde{u}|_{t=0} = \phi(x(y)), \label{E:28} \end{align} where the functions $\mathcal{A}^{\alpha\beta}_{kl}$ and $D^k$ satisfy conditions of the form \textbf{a), b),} and \textbf{c)} for $y\in B^+_1(0)$ but with the other parameters depending on the $\mathcal{C}^1$-norm of the diffeomorphism $y(x);\quad \psi(y)=\phi(x(y))\in L_{2,n-2+2\alpha}(B^+_1)$. The solution $\tilde{u}$ of (\ref{E:28}) satisfies the inequality (\ref{E:24}) with the other constants. Later on, we use the initial notation of the variables and functions, and prove our Theorem in the following local setting: \begin{align} u^k_t - \bigl(A^{\alpha\beta}_{kl}(z,u)u^l_{x_\beta}\bigr)_{x_\alpha} + b^k(z,u,u_x) = 0,\quad z\in Q^+=B^+_1(0)\times(0,T),\notag\\ u|_{\Gamma_1}=0,\quad \quad u|_{B^+_1(0)\times\{0\}}=\phi(x), \label{E:29} \end{align} and condition (\ref{E:26}) holds in a cylinder $Q_R(z^0)\subset Q^+$. We suppose that $\Gamma_R(z^0)=\mathbb{P}_R(z^0)\cap\Gamma_1\neq\emptyset$ and $\mathbb{P}_R(z^0)\cap\{t=0\}\neq\emptyset$. There is no loss of generality in assuming that \begin{equation} R^\alpha\leq\theta. \label{E:30} \end{equation} We will sharp the choice of $R\,$ later. \medskip \textbf{5}. \textit{Freezing coefficients in the problem.} \medskip We fix a point $\hat{z}\in \tilde{Q}_{\frac{3R}{4}}(z^0)= \overline{\Omega_{\frac{3R}{4}}(x^0)}\times\bigl[0,t^0+(\frac{3R}{4})^2\bigr]\supseteq Q_{\frac{3R}{4}}(z^0)$ and $r\leq \frac{R}{4a}$ (the constant $a\geq 2$ is fixed by (\ref{E:21}) ). We denote \begin{align*} Q_r(\hat{z})=Q_R(z^0)\cap \mathbb{P}_r(\hat{z}),\quad u_{r,\hat{z}}=\underset{Q_r(\hat{z})}{\oint}u(z)\,dz,\quad \hat{A}^{\alpha\beta}_{kl}=A^{\alpha\beta}_{kl}(\hat{z},u_{r,\hat{z}}) \end{align*} and consider the following problem: \begin{align} v^k_t - \hat{A}^{\alpha\beta}_{kl}v^l_{x_\beta x_\alpha}= 0\quad in\,\,Q_r(\hat{z}),\notag\\ v|_{\partial'Q_r(\hat{z})} = u(z). \label{E:31} \end{align} The problem has unique solution, it is smooth up to $\Gamma'_r(\hat{z})\bigcup\Omega^{(0)}_r(\hat{x})$, where $\Gamma'_r(\hat{z})=\partial'Q_r(\hat{z})\bigcap\Gamma_R(z^0)$ and $\Omega^{(0)}_r(\hat{x})=\partial'Q_r(\hat{z})\bigcap\{t=0\}$. ( The sets $\Gamma'_r(\hat{z})$ or $\Omega^{(0)}_r(\hat{x})$ can be empty.) First, we consider the case $\Omega^{(0)}_r(\hat{z})\neq\emptyset$. In this case, we introduce the function $\tilde{v}(z)=v(z)-\phi(x)$, it solves the problem \begin{align} \tilde{v}^k_t - \hat{A}^{\alpha\beta}_{kl}\tilde{v}^l_{x_\beta x_\alpha}= (\hat{A}^{\alpha\beta}_{kl}\phi^l_{x_\beta}(x))_{x_\alpha},\quad z\in Q_r(\hat{z})\notag\\ \tilde{v}|_{\Gamma'_r(\hat{z})\cup\Omega^{(0)}_r(\hat{x})}= 0. \label{E:32} \end{align} The following Campanato estimates are valid for $\tilde{v}$ ~\cite{G24}: \begin{align} \underset{Q_\rho(\xi)}{\int}|\tilde{v}-\tilde{v}_{\rho,\xi}|^2\,dz\leq c_{11}\Bigl[\Bigl(\frac{\rho}{\hat r}\Bigr)^{n+4} \underset{Q_{\hat{r}}(\xi)}{\int}|\tilde{v}-\tilde{v}_{\hat{r},\xi}|^2\,dz + c_{\phi}\hat{r}^{n+2+2\alpha}\Bigr],\label{E:33} \end{align} and \begin{align} \underset{Q_\rho(\xi)}{\int} |\tilde{v}_x|^2\,dz \leq c_{12}\Bigl[\Bigl(\frac{\rho}{\hat{r}}\Bigr)^{n+2}\underset{Q_{\hat{r}}(\xi)}{\int}|\tilde{v}_x|^2\,dz + c_\phi\hat{r}^{n+2\alpha}\Bigr],\quad \rho\leq\hat{r}\leq \frac {r}{2}, \label{E:34} \end{align} where $\xi\in \overline{\Omega_{\frac{r}{2}}(\hat{x})}\times[0,\hat{t}+(\frac{r}{2})^2]$ and $c_\phi=\|\phi_x\|_{L^{2,n-2+2\alpha}(\Omega^{(0)}_R(x^0))}.$ It is easy to see that estimates (\ref{E:33}) and (\ref{E:34}) imply the corresponding inequalities for $v$. Putting \begin{equation*} \Phi(\rho,\xi)=\underset{Q_\rho(\xi)}{\int}|v-v_{\rho,\xi}|^2\,dz, \end{equation*} we rewrite estimate (\ref{E:33}) in the form \begin{align} \Phi(\rho,\xi)\leq c_{13}\bigl[(\frac{\rho}{\hat{r}})^{n+4}\Phi(\hat{r},\xi) + c_\phi r^{2\alpha}\hat{r}^{n+2}\bigr],\quad \rho\leq\hat{r}\leq\frac{r}{2}. \label{E:35} \end{align} Due to the well-known Lemma by S.Campanato ~\cite{G24}, the inequality \begin{align} \Phi(\rho,\xi)\leq c_{14}\rho^{n+2}\bigl(\frac{\Phi(\hat{r},\xi)}{\hat{r}^{n+2}} + c_\phi r^{2\alpha}\bigr), \label{E:36} \end{align} $\rho\leq\hat{r}\leq\frac{r}{2},\,\,c_{14}=c_{14}(c_{13},n)$, follows from (\ref{E:35}). Now we put $\hat{r}=\frac{r}{2} $ in (\ref{E:36}) and obtain that \begin{align} \frac{\Phi(\rho,\xi)}{\rho^{n+2}}\leq c_{14}\Bigl\{\frac{\Phi(\frac{r}{2},\xi)}{(\frac{r}{2})^{n+2}} + c_\phi r^{2\alpha}\Bigr\} \leq \notag\\ c_{15}\bigl\{\frac{\Phi(r,\hat{z})}{r^{n+2}} + c_\phi r^{2\alpha}\bigr\}. \label{E:37} \end{align} We take the supremum over all admissible $\xi$ and $\rho\leq\frac{r}{2}$ to conclude that \begin{align} [v]^2_{\mathcal{L}^{2,n+2}(\tilde{Q}_{\frac{r}{2}}(\hat{z});\delta)} \leq c_{15}\Bigl\{\underset{Q_r(\hat{z})}{\oint}|v-v_{r,\hat{z}}|^2\,dz + c_\phi r^{2\alpha}\Bigr\} \label{E:38} \end{align} in the case $\Omega^{(0)}_r(\hat{x})\neq\emptyset$. We repeat the above considerations to deduce for $v_x$ from (\ref{E:34}) the estimate \begin{align} \|v_x\|^2_{L^{2,n}(\tilde{Q}_{\frac{r}{2}}(\hat{z}))}\leq c_{16}\Bigl\{\underset{Q_r(\hat{z})}{\dint}|v_x|^2\,dz + c_\phi r^{2\alpha}\Bigr\}, \label{E:39} \end{align} when $\Omega^{(0)}_r(\hat{x})\neq\emptyset$. If $\Omega^{(0)}_r(\hat{x})= \emptyset$, we do not transform problem (\ref{E:31}) to (\ref{E:32}), and obtain estimates of $[v]^2_{\mathcal{L}^{2,n+2}}(Q_{\frac{r}{2}}(\hat{z};\delta))$ and $\|v_x\|^2_{L^{2,n}(Q_{\frac{r}{2}}(\hat{z};\delta))}$ similar to (\ref{E:38}) and (\ref{E:39}) where the terms with $c_\phi$ are absent. Further, we consider function $w=u-v,\,w|_{\partial'Q_r(\hat{z})}=0$. It satisfies the identity \begin{align} \underset{Q_r(\hat{z})}{\int}(w^k_t h^k + \hat{A}^{\alpha\beta}_{kl}w^l_{x_\beta}h^k_{x_\alpha})\,dz=\underset{Q_r(\hat{z})}{\int} (F^k_\alpha(z)h^k_{x_\alpha}+f^k(z)h^k)\,dz,\notag\\ h\in L_2(\Lambda_r(\hat{t});\overset{0}{W}^1_2(\Omega_r(\hat{x})). \label{E:40} \end{align} Here $F^k_\alpha(z)=\hat{A}^{\alpha\beta}_{kl}u^l_{x_\beta}(z),\,\,f^k(z)=u^k_t(z).$ From (\ref{E:40}) with $h=w$, it follows that \begin{align} \underset{\Lambda_r(\hat{t})}{\sup}\|w(\cdot,t)\|^2_{2,\Omega(\hat{x})}+\|w_x\|^2_{2,Q_r(\hat{z})}\notag\\ \leq c_{17}(\|u_x\|^2_{2,Q_r(\hat{z})}+ r^2\|u_t\|^2_{2,Q_r(\hat{z})}). \label{E:41} \end{align} From (\ref{E:41}) and (\ref{E:24}), the estimate \begin{align} \underset{\Lambda_r(\hat{t})}{\sup}\|w(\cdot,t)\|^2_{2,\Omega_r(\hat{x})}+\|w_x\|^2_{2,Q_r(\hat{z})}\leq c_{18}\bigl(\|(1+|u_x|)\|^2_{2,Q_r(\hat{z})}\notag\\ +r^2\|\phi_x\|^2_{2,\Omega_r(\hat{x})}\bigr) \label{E:42} \end{align} follows. As a consequence of (\ref{E:38}),(\ref{E:25}),(\ref{E:42}), and (\ref{E:26}), we obtain the estimate \begin{align} [w]^2_{\mathcal{L}^{2,n+2}(\tilde{Q}_{\frac{r}{2}}(\hat{z}))}\leq c_{19}(\theta^2+r^{2\alpha})\underset{(30)}\leq 2c_{19}\theta^2, \label{E:43} \end{align} for the case $\Omega^{(0)}(\hat{x})\neq\emptyset.$ (The estimate of $[w]^2_{\mathcal{L}^{2,n+2}(Q_{\frac{r}{2}}(\hat{z}))}$ we have in the other case.) The inequality \begin{equation} \|w_x\|^2_{L^{2,n}(\tilde{Q}_{\frac{r}{2}}(\hat{z}))}\leq c_{20}\theta^2 \label{E:44} \end{equation} follows from (\ref{E:39}), (\ref{E:42}), and (\ref{E:26}), provided that $\Omega^{(0)}_R(\hat{x})\neq \emptyset$. If $\Omega^{(0)}_r(\hat{x})=\emptyset$, then the same estimate can be derived for $\|w_x\|^2_{L^{2,n}(Q_{\frac{r}{2}}(\hat{z}))}$. Besides (\ref{E:42}), we need a global $L_m$- estimate of $|w_x|$ in $Q_r(\hat{z})$ for an exponent $m>2$. Using identity (\ref{E:40}) and the condition $w|_{\partial'Q_r(\hat{z})}=0$, it is easy to deduce the reverse H\"{o}lder inequalities in all cylinders $Q_\rho\subset Q_r(\hat{z})$ (we admit $\partial Q_\rho\cap\partial'Q_r(\hat{z})\neq\emptyset$). By the parabolic version of the Gehring Lemma,we state that there exists a number $m=m(\nu,\mu,n) > 2$ such that \begin{align} \bigl(\underset{Q_r(\hat{z})}{\oint}|w_x|^m\,dz\bigr)^{\frac{2}{m}}\leq c_{21} \bigl\{\underset{Q_r(\hat{z})}{\oint}|w_x|^2\,dz + \bigl(\underset{Q_r(\hat{z})}{\oint}|F|^m\,dz\bigr)^{\frac{2}{m}} \notag\\ + r^2\bigl(\underset{Q_r(\hat{z})}{\oint}|f|^{\frac{lm}{2}}\,dz\bigr)^{\frac{4}{lm}}\bigr\}, \quad l=\frac{2(n+2)}{n+4}. \label{E:45} \end{align} We may consider (\ref{E:45}) with $m\leq \min\{p,\frac{2(n+4)}{n+2}\}$, where $p>2$ is the exponent from (\ref{E:21}). Such type estimate for $|w_x|$ but with the last term in the form \begin{equation*} \bigl(\underset{Q_r(\hat{z})}{\oint}|f|^{l+m-2}\, dz\bigr)^{\frac{2}{l+m-2}} \end{equation*} was obtained in ~\cite{A23}. Here we need the inequality (\ref{E:45}). One can deduce it, following the idea of proving Theorem 2.2 of Chapter 4 ~\cite{G24} for the standard euclidian metric. Now we estimate all terms in the right-hand side of (\ref{E:45}) in the way: \begin{align} \underset{Q_r(\hat{z})}{\oint}|w_x|^2\,dz \underset{(42)}\leq c_{22}\bigl(\underset{Q_r(\hat{z})}{\oint}(1+|u_x|)^2\,dz + \underset{\Omega_r^{(0)}(\hat{z})}{\oint}|\phi_x|^2\,dx\bigr), \notag \end{align} \begin{align} \bigl(\underset{Q_r(\hat{z})}{\oint}|F|^m\,dz\bigr)^{\frac{2}{m}}\leq c_{23}\bigl(\underset{Q_r(\hat{z})}{\oint}|u_x|^m\,dz\bigr)^{\frac{2}{m}}\,dz \notag\\ \underset{(21)}\leq c_{24}\underset{Q_{ar}(\hat{z})}{\oint}(1+|u_x|)^2\,dz + \underset{\Omega^{(0)}_{ar}(\hat{x})}{\oint}|\phi_x|^2\,dx,\notag \end{align} \begin{align} r^2\bigl(\underset{Q_r(\hat{z})}{\oint}|f|^{\frac{lm}{2}}\,dz\bigr)^{\frac{4}{lm}}\leq r^2\underset{Q_r(\hat{z})}{\oint}|u_t|^2\,dz.\notag \end{align} Now from (\ref{E:45}) and (\ref{E:24}), it follows that \begin{align} \bigl(\underset{Q_r(\hat{z})}{\oint}|w_x|^m\,dz\bigr)^{\frac{2}{m}} \leq c_{25} \Bigl[\underset{Q_{ar}(\hat{z})}{\oint}(1+|u_x|)^2\,dz + \underset{\Omega^{(0)}(\hat{x})}{\oint}|\phi_x|^2\,dx\Bigr]. \label{E:46} \end{align} Moreover, function $w$ satisfies the identity \begin{align} \underset{Q_r(\hat{z})}{\int}[w^k_t + \hat{A}^{\alpha\beta}_{kl}w^l_{x_\beta}\eta^k_{x_\alpha} + b^k(z,u,u_x)\eta^k + \Delta A^{\alpha\beta}_{kl}u^l_{x_\beta}\eta^k_{x_\alpha}]\,dz= 0,\notag\\ \eta|_{\Gamma_r(\hat{z})}=0, \label{E:47} \end{align} $\Delta A^{\alpha\beta}_{kl}=A^{\alpha\beta}_{kl}(z,u)-A^{\alpha\beta}_{kl}(\hat{z},u_{r,\hat{z}}).$ Further, we consider in detail the case $\Omega_r^{(0)}(\hat{x})\neq\emptyset$ and put in (\ref{E:47}) $\eta(z)=w(z)( (2T)^s -|w(z)|^s)_+,$ the parameters $T$ and $s\in(0,1)$ will be chosen later. Function $\eta$ is bounded in $Q_r(\hat{z})=\Omega_r\times(0,\hat{t}+r^2),\, (|\eta|\leq(2T)^{s+1})$, and $\eta|_{\partial'Q_r(\hat{z})}=0$. We obtain the inequality \begin{align*} \underset{Q_r}{\int}[w_tw((2T)^s-|w|^s)_+ + \hat{A}^{\alpha\beta}_{kl}w^l_{x_\beta}w^k_{x_\alpha}((2T)^s-|w|^s)_+\\ + \hat{A}^{\alpha\beta}_{kl}w^l_{x_\beta}w^k(-s|w|^{s-2}(w\cdot w_{x_\alpha})\chi_+)]\,dz \\ \leq \underset{Q_r}{\int}|\Delta A||u_x|(|w_x|((2T)^s-|w|^s)_+ + s|w|^s|w_x|\chi_+)\,dz \\ + b_0\underset{Q_r}{\int}|u_x|^2|w|((2T)^s-|w|^s)_+\,dz + \mu\underset{Q_r}{\int}|w|((2T)^s-|w|^s)_+\,dz,\quad Q_r=Q_r(\hat{z}), \end{align*} where $\chi_+(z)$ is the characteristic function of the set $\{z\in Q_r(\hat{z}): |w|<2T\}$. From this inequality follows that \begin{align} \underset{Q_r}{\int}\Bigl(\frac{|w|^2}{2}\Bigr)'_t((2T)^s-|w|^s)_+\,dz + \frac{\nu}{2}\underset{Q_r}{\int}|w_x|^2((2T)^s-|w|^s)_+\,dz \notag\\ \leq c_{26}T^s \bigl\{s\underset{Q_r}{\int}|w_x|^2\,dz + \underset{Q_r}{\int}\omega^2(r^2;|u-u_{r,\hat{z}}|^2)|u_x|^2\,dz \notag\\+ b_0\underset{Q_r}{\int}|u_x|^2|w|\chi_+\,dz + \underset{Q_r}{\int}|w|\chi_+\,dz\bigr\} \label{E:48} \end{align} The first integral in the left-hand side of (\ref{E:48}) is nonnegative. Indeed, \begin{align} J=\underset{Q_r}{\int}\Bigl(\frac{|w|^2}{2}\Bigr)'_t((2T)^s-|w|^s)_+\,dz= \underset{\Omega_r(\hat{x})}{\int}\Bigl(\underset{0}{\int}^{|w(x,t)|}\xi((2T)^s-\xi^s)_+ \,d\xi \Bigr)\,dx\Bigr|_{t=0}^{t=\hat{t}+ r^2} \notag\\ = \underset{\Omega_r(\hat{x})}{\int}\Bigl( \underset{0}{\int}^{|w(x,\hat{t}+ r^2)|}\xi((2T)^s-\xi^s)_+\,d\xi\Bigr)\,dx\geq 0. \label{E:49} \end{align} Now we'll estimate from below the second integral in the left-hand side of (\ref{E:48}). First,we fix the number \begin{equation} k_\ast=\frac{T}{2^\frac{1}{s}}, \label{E:50} \end{equation} and put \begin{align*} q=\frac{1}{2\sqrt{n}},\quad\tilde{Q}_{\rho}(\hat{z})=\Omega_\rho(\hat{x})\times(0,\hat{t}+\rho^2) \supseteq Q_\rho(\hat{z})=\mathbb P_\rho(\hat{z})\cap Q^+, \end{align*} \begin{align*} \tilde{Q}^+_{qr}(\hat{z})=\{z\in\tilde{Q}_{qr}(\hat{z}):\,\,|w(z)-w_{qr,\hat{z}}|>k_\ast\} \\ \tilde{Q}^-_{qr}(\hat{z})=\{z\in\tilde{Q}_{qr}(\hat{z}):\,\,|w(z)-w_{qr,\hat{z}}|\leq k_\ast\}. \end{align*} Note, that \begin{align*} |w_{qr,\hat{z}}| \leq\frac{|Q_r|}{|Q_{qr}|}\underset{Q_r(\hat{z})}{\oint}|w|\,dz = c_{27}\underset{Q_r(\hat{z})}{\oint}|w|\,dz\leq c_{28}\bigl(\underset{Q_r(\hat{z})}{\dint}|w_x|^2\,dz\bigr)^{\frac{1}{2}}\\ \underset{(42)}{\leq} c_{29}\Bigl(\underset{Q_{2r}(\hat{z})}{\dint}(1+|u_x|)^2\,dz + \underset{\Omega_{2r}^{(0)}(\hat{x})}{\dint}|\phi_x|^2\,dx\Bigr)^{\frac{1}{2}}\underset{(26)}{\leq} c_{30}(\theta + r^\alpha)\underset{(30)}{\leq}c_{31}\theta. \end{align*} \noindent We assume that the parameters $\theta \leq 1$ , and get the inequality \begin{equation} |w_{qr,\hat{z}}|\leq c_{31}. \label{E:51} \end{equation} Let the numbers $T$ and $s$ be fixed to satisfy the condition \begin{align} k_\ast = \frac{T}{2^{\frac{1}{s}}} > c_{31}. \label{E:52} \end{align} It yields the estimate \begin{align*} |w(z)|\leq |w(z)-w_{qr,\hat{z}}| + |w_{qr,\hat{z}}|\leq 2k_\ast,\quad z\in\tilde{Q}^-_{qr}(\hat{z}). \end{align*} Consequently, \begin{align} (2T)^s -|w(z)|^s\geq (2T)^s -(2k_\ast)^s= \frac{(2T)^s}{2} \label{E:53} \end{align} on the set $\tilde{Q}^-_{qr}(\hat{z}).$ From the said follows that \begin{align} \mathcal{L}=\underset{Q_r}{\int}|w_x|^2((2T)^s-|w|^s)_+\,dz \geq \underset{\tilde{Q}_{qr}^-(\hat{z})}{\int}|w_x|^2((2T)^s-|w|^s)_+\,dz \notag\\ \geq \frac{(2T)^s}{2}\underset{\tilde{Q}_{qr}^-(\hat{z})}{\int}|w_x|^2\,dz. \label{E:54} \end{align} Taking into account the estimates for $J$ and $\mathcal{L}$ and dividing inequality (\ref{E:48}) by $(2T)^s$, we derive the relation \begin{align} \underset{\tilde{Q}^-_{qr}(\hat{z})}{\int}|w_x|^2\,dz \leq c_{32}\Bigl\{s\underset{Q_r(\hat{z})}{\int}|w_x|^2\,dz + \underset{Q_r(\hat{z})}{\int}\omega^2|u_x|^2\,dz \notag\\ + \underset{Q_r(\hat{z})}{\int}|u_x|^2|w|\chi_+\,dz + \underset{Q_r(\hat{z})}{\int}|w|\,dz \Bigr\}. \label{E:55} \end{align} Now we explain the estimating of every integral in the right-hand side of (\ref{E:55}). The first term will be estimated by (\ref{E:42}). The second integral we estimate by the standard way with the help of (\ref{E:21}): \begin{align} \underset{Q_r(\hat{z})}{\int}\omega^2(r^2;|u-u_{r,\hat{z}}|^2)|u_x|^2\,dz\leq \bigl(\underset{Q_r(\hat{z})}{\oint}|u_x|^p\,dz\bigr)^{\frac{2}{p}} \bigl(\underset{Q_r(\hat{z})}{\oint}\omega^{\frac{2p}{p-2}}\,dz\bigr)^{\frac{p-2}{p}}|Q_r| \notag\\ \underset{(21)}\leq c_{33}\Bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + r^2\underset{\Omega^{(0)}_{ar}(\hat{x})}{\int}|\phi_x|^2\,dx\Bigr) \omega^{\frac{p-2}{p}}(r^2;\underset{Q_r}{\oint}|u-u_{r,\hat{z}}|^2\,dz)\notag\\ \underset{(25)}\leq c_{33} \Bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + r^2\underset{\Omega_{ar}^{(0)}(\hat{x})}{\int}|\phi_x|^2\,dx\Bigr)\notag\\ \cdot\omega^{\frac{p-2}{p}}\bigl(r^2; c_9\bigl(\underset{Q_{2r}}{\dint}(1+|u_x|)^2\,dz + \underset{\Omega_{2r}^{(0)}(\hat{x})}{\dint}|\phi_x|^2\,dx\bigr)\bigr) \label{E:56} \end{align} Further, \begin{align} \underset{Q_r(\hat{z})}{\int}|u_x|^2|w|\chi_+\,dz\leq\Bigl(\underset{Q_r(\hat{z})}{\oint}|u_x|^p\,dz\Bigr)^{\frac{2}{p}} \Bigl(\underset{Q_r(\hat{z})}{\oint}|w|^{\frac{p}{p-2}}\chi_+\,dz\Bigr)^{1-\frac{2}{p}}|Q_r|\notag\\ \underset{(21)}{\leq} c_{34} \Bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + r^2\underset{\Omega^{(0)}_{ar}(\hat{x})}{\int}|\phi_x|^2\,dx\Bigr) T^{\frac{4-p}{p}}\Bigl(\underset{Q_r(\hat{z})}{\oint}|w|^2\,dz\Bigr)^{\frac{p-2}{p}}\notag\\ \underset{(42),(26)}{\leq} c_{35} T^{\frac{4-p}{p}}(\theta^2+r^2)^{\frac{p-2}{p}}\Bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + c_\phi r^{n+2\alpha}\Bigr). \label{E:57} \end{align} At last, \begin{align} \underset{Q_r(\hat{z})}{\int}|w|\,dz \leq c(n) (r^2\underset{Q_r(\hat{z})}{\int}|w_x|^2\,dz + r^{n+2})\notag\\ \underset{(42)}\leq c_{36} r^2 \underset{Q_{2r}(\hat{z})}{\int}(1+|u_x|)^2\,dz + c_{37}r^{n+2}. \label{E:58} \end{align} Using (\ref{E:56}) --(\ref{E:58}), we derive from (\ref{E:55}) the inequality \begin{align} \underset{\tilde{Q}_{qr}(\hat{z})}{\int}|w_x|^2\,dz \leq \underset{\tilde{Q}_{qr}^+(\hat{z})}{\int}|w_x|^2\,dz + c_{38}\Bigl\{s+\omega^{\frac{p-2}{p}}(r^2; c_{39}\theta^2) \notag\\ + T^{\frac{4-p}{p}}\theta^{\frac{2(p-2)}{p}} + r^2\Bigr\} \Bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + c_\phi r^{n+2\alpha}\Bigr). \label{E:59} \end{align} Now we estimate integral $\mathcal{M} = \underset{\tilde{Q}^+_{qr}(\hat{z})}{\int}|w_x|^2\,dz $ in (\ref{E:59}): \begin{align} \mathcal{M}\leq \bigl(\underset{\tilde{Q}_{qr}(\hat{z})}{\oint}|w_x|^m\,dz\bigr)^{\frac{2}{m}}|\tilde{Q}^+_{qr}|^{1-\frac{2}{m}}|\tilde{Q}_{qr}|^{\frac{2}{m}}\notag\\ \underset{(46),(42)}{\leq } c_{39}\Bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + c_\phi r^{n+2\alpha}\Bigr)\Bigl(\frac{|\tilde{Q}^+_{qr}|}{|\tilde{Q}_{qr}|}\Bigr)^{1-\frac{2}{m}}. \label{E:60} \end{align} The next step is to explain that the relation $\frac{|\tilde{Q}^+_{qr}|}{|\tilde{Q}_{qr}|}$ is a decreasing function with respect to the parameter $k_\ast$ (see (\ref{E:50})). We consider cube $D_r(\hat{z})=\{(x,t):\, |x_i-\hat{x}_i|< r,\,i\leq n,\,\,|t-\hat{t}|< r^2\}$ and put \begin{align*} w_0(z)=\begin{cases} w(z),\quad z\in Q_r(\hat{z}); \\ 0, \quad z\in D_r(\hat{z})\setminus Q_r(\hat{z}). \end{cases} \end{align*} We assert that \begin{align} \tilde{Q}^+_{qr}(\hat{z})\subset\{z\in D_{qr}(\hat{z}):\,|w^0(z)-w^0_{qr,\hat{z}}| >\frac{k_\ast}{2}\}\equiv D^{(k_\ast)}_{qr}(\hat{z}),\notag\\ w^0_{qr,\hat{z}} = \frac{1}{|D_{qr}|}\underset{D_{qr}(\hat{z})}{\int}w^0\,dz. \label{E:61} \end{align} Indeed, let $z\in\tilde{Q}_{qr}(\hat{z})$ and $|w(z)-w_{qr,\hat{z}}| > k_\ast\,$ then \begin{align} |w^0(z)-w^0_{qr,\hat{z}}| = |w(z)-w^0_{qr,\hat{z}}|\geq \bigl||w(z)-w_{qr,\hat{z}}| - |w_{qr,\hat{z}} - w^0_{qr,\hat{z}}|\bigr|\equiv{l}.\notag \end{align} Looking at the derivation of (\ref{E:50}), one can see that \begin{align} |w^0_{qr,\hat{z}}|\leq \frac{|Q_r|}{|D_{qr}|}\underset{Q_r(\hat{z})}{\oint}|w|\,dz\leq c_{27}\underset{Q_r(\hat{z})}{\oint}|w|\,dz\leq\dots\leq c_{31}\theta\underset{(52)}{\leq}k_\ast. \notag \end{align} We make stronger condition (\ref{E:52}) and assume that \begin{equation} k_\ast > 4c_{31}. \label{E:62} \end{equation} Then $|w_{qr,\hat{z}} - w^0_{qr,\hat{z}}|\leq 2c_{31}\leq \frac{k_\ast}{2}$ and $l> k_\ast -\frac{k_\ast}{2} = \frac{k_\ast}{2}$. Imbedding (\ref{E:61}) follows. Moreover, it is easy to check that \begin{align} [w^0]_{\mathcal{L}^{2,n+2}(D_{qr}(\hat{z});\delta)} \leq c_{40} \bigl([w]_{\mathcal{L}^{2,n+2}(\tilde{Q}_{\frac{r}{2}}(\hat{z};\delta))}\notag\\ + \|w_x\|_{L^{2,n}(\tilde{Q}_{\frac{r}{2}};\delta)}\bigr) \underset{(43),(44)}{\leq} c_{41}\theta\leq c_{41}. \label{E:63} \end{align} Due to the parabolic version of the John-Nirenberg theorem, we can assert that there exist positive numbers $H$ and $\beta$ such that \begin{align} \frac{|\{z\in D_R(\hat{z}):\,|g(z)-g_{R,\hat{z}}|>\lambda\}|}{|D_R|} \leq H\, exp\Bigl(\frac{-\beta\lambda}{[g]_{\mathcal{L}^{2,n+2}(D_R;\delta)}}\Bigr), \label{E:64} \end{align} for a function $g\in\mathcal{L}^{2,n+2}(D_R; \delta), \quad \lambda>0$. The numbers $H$ and $\beta$ in (\ref{E:64}) depend on the dimension $n $ only. This fact was proved for the standard euclidian metric in ~\cite{J-N25} and can be generalized for the parabolic metric in the same way. We put $g(z)=w^0(z),\,R=qr,\lambda=\frac{k_\ast}{2},$ and obtain the estimate \begin{align} \frac{|\tilde{Q}^+_{qr}(\hat{z})|}{|\tilde{Q}_{qr}|} \underset{(61)}{\leq} c(n)\frac{|D^{(k_\ast)}_{qr}(\hat{z})|}{|D_{qr}|} \underset{(64)}{\leq} c(n)H \exp\Bigl(\frac{-\beta k_\ast}{2[w^0]_{\mathcal{L}^{2,n+2}(D_{qr}; \delta)}}\Bigr). \label{E:65} \end{align} Using (\ref{E:63}) and (\ref{E:65}), we derive from (\ref{E:60}) the inequality \begin{align} \mathcal{M}\leq c_{42} \exp(-c_{43}k_\ast)\bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + r^{n+2\alpha}\bigr). \label{E:66} \end{align} Now it follows from (\ref{E:59}) and (\ref{E:66}) that \begin{align} \underset{\tilde{Q}_{qr}(\hat{z})}{\int}|w_x|^2\,dz \leq c_{44}\Bigl\{\exp(-c_{43}k_\ast) + s + \omega^{\frac{p-2}{p}}(r^2; c_{39}\theta^2) \notag\\ + T^{\frac{4-p}{p}}\theta^{\frac{2(p-2)}{p}} + r^2 \Bigr\}\Bigl(\underset{Q_{ar}(\hat{z})}{\int}(1+|u_x|)^2\,dz + c_\phi r^{n+2\alpha}\Bigr). \label{E:67} \end{align} We put $\psi(\rho,\hat{z})=\underset{Q_\rho(\hat{z})}{\int}(1+|u_x|)^2\,dz$ and consider (\ref{E:34}) for $v$ (with $\xi=\hat{z},\,\hat{r}=qr$), and (\ref{E:67}) to obtain the estimate \begin{align} \psi(\rho,\hat{z})\leq c_{45}\Bigl\{ \bigl(\frac{\rho}{r}\bigr)^{n+2} + \bigl[s + exp(-c_{43}\frac{T}{2^1/s}) + \omega^{\frac{p-2}{p}}(r^2; c_{39}\theta^2) \notag\\+ T^{\frac{4-p}{p}}\theta^{\frac{2(p-2)}{p}} + r^2\bigr]\Bigr\} \psi(ar,\hat{z}) + c_{46}(1+T^{\frac{4-p}{p}}\theta^{\frac{2(p-2)}{p}})r^{n+2\alpha},\notag\\ \rho\leq qr. \label{E:68} \end{align} Obviously, inequality (\ref{E:68}) is valid also for $\rho\in (qr,\,ar].$ Further we change the notation $ar$ by $r$, and consider $r\leq \frac{R}{4}.$ It gives the inequality \begin{align} \psi\bigl(\rho,\hat{z}\bigr)\leq c_{47}\Bigl\{(\frac{\rho}{r})^{n+2} + [...]\Bigr\}\psi(r,\hat{z}) + c_{48} (1+T^{\frac{4-p}{p}}\theta^{\frac{2(p-2)}{p}})r^{n+2\alpha},\notag\\ \rho\leq r. \label{E:69} \end{align} In (\ref{E:69}) the expression $[...]$ coincides with the brackets of (\ref{E:68}). By the well-known Campanato Lemma (see, for example, ~\cite{G24}, Chapter 2, Lemma 2.1), there exists a number $\delta_0=\delta_0(c_{47},n,\alpha) >0$ such that if \begin{equation} [\cdots] < \delta_0 \label{E:70} \end{equation} in inequality (\ref{E:69}), then \begin{align} \psi(\rho,\hat{z}) \leq c_{49} \bigl(\frac{\rho}{r}\bigr)^{n+2\alpha}\{\psi(r,\hat{z}) + (1+T^{\frac{4-p}{p}}\theta^{\frac{2(p-2)}{p}})r^{n+2\alpha}\},\quad \rho\leq r. \label{E:71} \end{align} Here $c_{49}$ depends on $c_{47}, n,$ and $\alpha$. At last, we fix the parameters $s,T,\theta$ and $R$. First, let $s$ be fixed in $(0,1)$ to satisfy \begin{align} s < \frac{\delta_0}{4}. \label{E:72} \end{align} Then, we fix $T>1$ to obtain (\ref{E:62}) and the inequality \begin{align} \exp(-c_{43}\frac{T}{2^1/s}) <\frac{\delta_0}{4}. \label{E:73} \end{align} At last, we choose numbers $\theta_1$ and $R_1$ to obtain the inequality \begin{align} T^{\frac{4-p}{p}}\theta_1^{\frac{2(p-2)}{p}} + \omega^{\frac{p-2}{p}}(R_1; c_{39}\theta^2_1) + R_1^2 < \frac{\delta_0}{2} \label{E:74} \end{align} Now we may fix $R\leq 4R_1$ and $ \theta\leq \theta_1$ in assumption (\ref{E:26}). Conditions (\ref{E:72})--(\ref{E:74}) guarantee the validity of (\ref{E:70}). Under restrictions $r\leq\frac{R}{4}\leq R_1$ and $\theta\leq\theta_1$, inequality (\ref{E:71}) follows. We obtain the relation \begin{align} \frac{1}{\rho^{n+2\alpha}}\underset{\tilde{Q}_\rho(\hat{z})}{\int}|u_x|^2\,dz \leq c_{50}\bigl\{R^{-(n+2\alpha)}\underset{Q_R(z^0)}{\int}|u_x|^2\,dz + 1 \bigr\}, \label{E:75} \end{align} for all $\hat{z} \in \tilde{Q}_{\frac{3R}{4}}(z^0),\,\rho\leq\frac{R}{4}.$ It was considered the case $\Omega^{(0)}_r(\hat{x})\neq\emptyset$. Now we address to the situation $\Omega^{(0)}_r(\hat{x}) = \emptyset$. It should be noted only that we do not transform problem (\ref{E:31}) to (\ref{E:32}) in this case, and apply estimates (\ref{E:33}) and (\ref{E:34}) to the function $v$. Moreover, now estimates (\ref{E:35}) -- (\ref{E:36}) and others do not include the terms with $c_\phi$. One can repeat all considerations and assert that estimate (\ref{E:75}) is valid for all $\hat{z}\in Q_{\frac{3R}{4}}(z^0),\,\rho\leq \frac{R}{4},\,$ provided that $\Omega^{(0)}(\hat{x})=\emptyset$. Inequalities (\ref{E:25}) and (\ref{E:75}) ensure the estimate \begin{align} [u]^2_{\mathcal{L}^{,n+2+2\alpha}(Q_{\frac{3R}{4}}(z^0); \delta)} \leq c_{51}\{R^ {-(n+2\alpha)}\|u_x\|^2_{2,Q_R(z^0)} +1\}. \label{E:76} \end{align} Due to the isomorphism of $\mathcal{L}^{2,n+2+2\alpha}(Q;\delta)$ to the H\"{o}lder space, estimate (\ref{E:76}) yields the estimate of the seminorm in the H\"{o}lder space \begin{align} ^{(\alpha,\alpha/2)}_{Q_{\frac{3R}{4}}(z^0)} \leq c_{52}\{R^{-(n+2\alpha)}\|u_x\|^2_{2,Q_R(z^0)} + 1\}. \label{E:77} \end{align} We have proved the Theorem for the situation $\Omega^{(0)}_R(x^0)\neq\emptyset$ and $\Gamma_R(z^0)\neq\emptyset$. In the other cases, all steps of the proof are only simplified. $\blacksquare$ \bigskip This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no.05-01-01063). \medskip \begin{thebibliography}{28} \bibitem{AA1}A.Arkhipova. 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