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\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|<R\},\notag\\
&\Omega_R(x^0)=\Omega\cap B_R(x^0),\quad
T_R(x^0)=\{x\in\mathbb{R}^n: \,|x_i-x^0_i|<R,\,i=1,...,n\},\notag\\
&\Lambda_R(t^0)=(t^0-R^2,t^0+R^2),\,\,
\mathbb{P}_R(z^0)=B_R(x^0)\times\Lambda_R(t^0),\notag \\
& Q_R(z^0)=\mathbb{P}_R(z^0)\cap Q,\,\,
\Gamma_R(z^0)=\mathbb{P}_R(z^0)\cap\Gamma,\,\,\Omega^{(0)}(x^0)=\mathbb{P}_R(z^0)\cap\{t=0\},\notag\\
\end{align*}
 $\partial'Q_R(z^0)\,$  is the parabolic boundary of $Q_R(z^0)$,
 $|D|= meas_{n+1}D\,$\, for  a Lebesgue measurable set in
$\mathbb{R}^{n+1}$,  $d_0=diam\,\Omega$,
\begin{align*}
g_{r,z^0}=\underset{Q_r(z^0)}{\oint}
g\,dz=\frac{1}{|Q_r|}\underset{Q_r(z^0)}{\int}g \,dz,\quad
|Q_r|=2\omega_nr^{n+2},\,\,
\omega_n=meas_n B_1(0), \notag\\
\underset{Q_r(z^0)}{\dint}g\,dz=\frac{1}{r^n}\underset{Q_r(z^0)}{\int}g\,dz,\quad
 \underset{\Omega_r(x^0)}{\dint}\psi\,dx=\frac{1}{r^{n-2}}\underset{\Omega_r(x^0)}
 {\int}\psi\,dx,\notag\\
 \end{align*}
 $\|u\|_{m,D}$ is the norm of a function $u$ in the space
 $L_m(D),\,m\in[1,\infty]$,
\medskip

 \noindent
 $\delta(z^1,z^2)=\max{\{|x^1-x^2|,|t^1-t^2|^\frac{1}{2}\}}$,\quad
 $z^i=(x^i,t^i)\in \mathbb{R}^{n+1},\,i=1,2.$
 \medskip

\noindent $\bigl< u \bigr>^{(\alpha,\frac{\alpha}{2})}_Q =
\underset{z,z'\in\overline{Q},\,z\neq
z'}{\sup}\frac{|u(z)-u(z')|}{\delta(z,z')^\alpha}.$
\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}
<u>^{(\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. "Local and global solvability of the
Cauchy-Dirichlet problem for nondiagonal parabolic systems in the
case of two spatial variables" / Amer.Math.Soc.Transl.
\textbf{209}, n.2, (2000),1-19; in Russian: Algebra \&
Analysis,\textbf{11}, n.6, (1999), 69-102.

\bibitem{AA2}A.Arkhipova. "Continuability in time of smooth
solutions of strong-nonlinear nondiagonal parabolic systems"/
Ann.Scuola Norm Sup.,Pisa, Cl.Sci.(5),\textbf{1}, (2002), 153-167.

\bibitem{AA3}A.Arkhipova. "Global solvability of the
Cauchy-Dirichlet problem for nondiagonal parabolic systems with
variational structure in the case of two spatial variables." / J.
Math.Sci.,\textbf{92}, (1998),4231-4255; in Russian: Probl.Mat.
Anal.,St.Petersburg Univ.,\textbf{16}, (1997), 3-40.

\bibitem{AA4} A.Arkhipova. "Cauchy-Neumann problem for a class of
nondiagonal parabolic systems with quadratic growth
nonlinearities. I. On the continuability of smooth solutions"./
Comment.Math.Univ.Carolin., \textbf{41}, n.4, (2000), 693-718.

\bibitem{AA5} A.Arkhipova. "Cauchy-Neumann problem for a class of
nondiagonal parabolic systems with quadratic growth
nonlinearities. II. Local and global solvability results."./
Comment.Math.Univ.Carolin., \textbf{42}, n.1, (2001), 53-76.

\bibitem{Ch-Str6} Y.Chen,,M.Struwe. "Existence and partial
regularity results for the heat flow of harmonic maps"./
Math.Z.,\textbf{201} (1989), 83-103.

\bibitem{Str7} M.Struwe. "On the evolution of harmonic mappings of
Riemannian surfaces"./ Comment.Math.Helv.,\textbf{60}, (1985),
558-581.

\bibitem{J-St8} J.Stara, O.John. "Some (new) counterexamples of
parabolic systems"./Comment.Math.Univ.Carolin.,\textbf{36}, n.3,
(1995), 503-510.

\bibitem{St-J-M9} J.Stara, O.John, J.Mal\`{y}. "Counterexamples to
the regularity of weak solution of the quasilinear parabolic
system." / Comment.Math.Univ.Carolin.,\textbf{27} (1986), 123-126.

\bibitem{L-S-U10} O.Ladyzhenskaya,V.Solonnikov, N.Uraltseva. Linear
and Quasilinear Equations of Parabolic Type./ Transl.of Math.
Monographs 23. Providence, Rhode Island, AMS (1968).

\bibitem{T11} P.Tolksdorf. "On some parabolic variational problems
with quadratic growth." / Ann.Scuola Norm.Super.,Pisa,
\textbf{13}, (1986), 193-223.

\bibitem{W12} W.Wieser. "On parabolic systems with variational
structure"./Manuscripta Math. \textbf{54}, (1985), 53-82.

\bibitem{Fr13} J.Frehse. "On two-dimensional quasilinear elliptic
systems." /Manuscripta Math.,\textbf{28}, (1979), 21-50.

\bibitem{G-Str14} M.Giaquinta, M.Struwe. "On the partial regularity
of weak solutions of nonlinear parabolic systems." /Math.Z.,
\textbf{179}, (1982), 437-451.

\bibitem{M-M15} M.Marino,A.Maugeri. "Partial H\"{o}lder continuity of
solutions of nonlinear parabolic systems of second order with
quadratic growth."/ Bolletino U.M.I.,\textbf{3-B} (1989), 397-435.

\bibitem{M-M16} M.Marino, A.Maugeri./ "A remark on the
Note: Partial H\"{o}lder continuity of the spatial derivatives of
the solutions to nonlinear parabolic systems with quadratic
growth". / Rendic.Sem.Mat., Padova, \textbf{95}, (1996).

\bibitem{A17} A.Arkhipova."On the partial regularity up to the
boundary of weak solutions to quasilinear parabolic systems with
quadratic growth." / Zapisky Nauchn.Seminar. POMI, St-Petersburg,
\textbf{249}, (1997), 3-23.

\bibitem{A18} A.Arkhipova. "Quasireverse H\"{o}lder inequalities and a
priori estimates for quasilinear elliptic systems".
/Rendic.Mat.Acc.Lincei, \textbf{14},n.2, Serie 9, (2003), 91-108.

\bibitem{A19} A.Arkhipova. "Boundary a priori estimates for
solutions of nondiaginal elliptic systems with strong
nonlinearities". / Russian Avad.Izvestia Math., \textbf{68}, n.2,
(2004), 243-258.

\bibitem{A20} A.Arkhipova. The monotonicity condition and a priori
estimates of the H\"{o}lder norm for a class of nondiagonal
elliptic systems with quadratic nonlinearity". / J. of Math.Sci.,
\textbf{142}, n.1, (2007), 1733-1748; in Russian: Problemy Matem.
Analiza, St-Petersburg, \textbf{34}, (2006), 3-13.

\bibitem{A21} A.Arkhipova. "Quasireverse H\"{o}lder inequalities in
parabolic metric and their applications"./ Amer.Math.Soc.Transl.
(2), to appear in 2007.

\bibitem{A22} A.Arkhipova. "L-p estimates of the gradients of
solutions of the initial-boundary problems for quasilinear
parabolic systems". /in Russian: Problemy Nat.Anzliza,
\textbf{13}, St.Petersburg Univ.,(1992), 5-18.

\bibitem{A23} A.Arkhipova. "On the regularity of the solution
 of the Neumann problem for quasilinear parabolic systems".
 /Russian Acad. Izvestia Math., \textbf{45}, n.2, (1995), 231-253;
 in Russian: Izvestia RAN, ser. Math.,\textbf{58}, n.5,
 (1994),3-25.

 \bibitem{G24} M.Giaquinta. Multiple Integrals in the Calculus of
 Variations and Nonlinear Elliptic Systems. / Annals of Math.,
 Studies 105, Princeton Univ.Press, Princeton, (1983).

 \bibitem{J-N25} A.Kufner, O.John, S.Fuc\v{i}k. Functional Spaces. /
 Academia, Praha, (1977).

 \bibitem{G-G26} M.Giaquinta, E.Giusti. "Nonlinear elliptic systems
 with quadratic growth". / Manuscripta Math. \textbf{24}, (1978),
 323-349.

 \bibitem{A28} A.Arkhipova. "New a priori estimates for
 q-nonlinear elliptic systems with strong nonlinearities in the
 gradient, $1<q<2$". /Zapisky Nauchn.Semin. POMI, \textbf{310},
 (2003), 19-48.

 \bibitem{A29} A.Arkhipova. "Reverse H\"{o}lder inequalities with
 boundary integrals and L-p estimates for solutions of nonlinear
 elliptic and parabolic boundary-value problems". / Amer. Math.
 Soc. Transl. \textbf{164}, n.2, (1995), 15-42.
\end{thebibliography}
\bigskip
\begin{abstract}
We consider nondiagonal elliptic and parabolic systems of
equations with quadratic nonlinearities in the gradient. We
discuss new description of regular points of solutions of such
type systems. For a class of strongly nonlinear parabolic systems,
 we estimate locally the H\"older norm of a solution. Instead of
smallness of the oscillation , we assume local smallness of the
Campanato seminorm of the solution under consideration. Theorems
about \textit{quasireverse} H\"older inequalities proved by the
author are essentially used. We study systems under the Dirichlet
boundary condition and estimate the H\"older norm of a solution up
to the boundary ( up to the parabolic boundary of the prescribed
cylinder in the parabolic case).
\end{abstract}

\end{document}
