\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 57, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2007/57\hfil Nonexistence of the gradient blow-up phenomenon]
{Sufficient conditions for nonexistence of gradient
 blow-up for  nonlinear parabolic equations}

\author[A. S. Tersenov\hfil EJDE-2007/57\hfilneg]
{Aris S. Tersenov}

\address{Aris S. Tersenov \newline
Department of Mathematics and Statistics \\
University of Cyprus \\
P.O. Box 20537 \\
1678 Nicosia, Cyprus \newline
Tel.:+357 22892560, Fax:+357 22892550}
 \email{aterseno@ucy.ac.cy}

\thanks{Submitted February 8, 2006. Published April 17, 2007.}
\subjclass[2000]{35K55, 35K15, 35A05}
\keywords{Bernstein-Nagumo condition; gradient blow-up;
 a priori estimates \hfill\break\indent
nonlinear parabolic equation}

\begin{abstract}
 In this paper we study the initial-boundary value problems for
 nonlinear parabolic equations without Bernstein-Nagumo condition.
 Sufficient conditions guaranteeing the nonexistence of gradient
 blow-up  are formulated. In particular, we show that for a wide
 class of nonlinearities the Lipschitz continuity in the space
 variable together with the strict monotonicity with respect to the
 solution guarantee that gradient blow-up cannot occur at the
 boundary or in the interior of the domain.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In the present paper we consider the nonlinear equation
\begin{equation}\label{1.1}
u_t=F(t,x,u,u_x,u_{xx}) \quad \text{in } Q_T=(-l,l)\times (0,T),
\end{equation}
coupled with one of the boundary conditions
\begin{gather}\label{1.2}
u_x(t,-l)=u_x(t,l)=0,
\\ \label{1.3}
u_x+\sigma_{1}(t,x,u)\big|_{x=-l}= u_x+\sigma_{2}(t,x,u)\big|_{x=l}=0,
\\ \label{1.4}
u(t,-l)=u(t,l)=0
\end{gather}
and the initial condition
\begin{equation}\label{1.5}
u(0,x)=u_0(x).
\end{equation}
We assume that $F(t,x,u,p,r)$ is continuously differentiable with
respect to $r$ and satisfies the parabolicity condition, i.e.
\begin{equation}\label{i.1}
F_r(t,x,u,p,r)>0 \quad \text{for} \quad (t,x,u,p,r)
\in \overline{Q}_T\times [- M,M]\times \mathbb{R}^2.
\end{equation}
Write equation \eqref{1.1} in the  form
\begin{equation}\label{i.2}
u_t=F_r(t,x,u,u_x,\lambda u_{xx})u_{xx}+F(t,x,u,u_x,0),\quad \lambda\in [0,1],
\end{equation}
using the mean value theorem. The well known Bernstein-Nagumo
condition \cite{BERN, SNB, Nag} (see also \cite{ACONST, SNK, KRYL,
LSV, LIB, SER}) in the case of equation \eqref{i.2} appears as
\begin{equation}\label{i.3}
\frac{|F(t,x,u,p,0)|}{F_r(t,x,u,p,r)}\leq \phi (|p|) \quad \text{for}
\quad (t,x,u,p,r)\in \overline{Q}_T\times [- M,M]\times \mathbb{R}^2,
\end{equation}
where $\phi(\rho)$ is a nondecreasing positive function such that
$$
\int^{+\infty}\frac {\rho d\rho}{\phi (\rho)} =  + \infty.
$$
Condition \eqref{i.3} guarantees global a priori estimate for the
gradient of  bounded solutions. There are examples showing that a
violation of the Bernstein-Nagumo condition can imply the gradient
blow-up on the boundary as well as at interior points of the domain
(see \cite{ANFI, ASISH, DL, Y, KUT, SOUPL, VZ}), while the solution itself
remains bounded. Recently, in \cite{TT}, condition \eqref{i.3} was
substituted by a less restrictive one that allows an arbitrary
growth of $F(t,x,u,p,0)$ with respect to $p$ (see also
\cite{souplet}).

Let us recall some of the main results that were established in
\cite{TT}. Suppose that the right hand side of equation
\eqref{i.2} can be represented as follows
\begin{equation}\label{i.4}
F(t,x,u,p,0)=f_1(t,x,u,p)+f_2(t,x,u,p),
\end{equation}
where $f_2$ satisfies the restrictions
\begin{gather}\label{i.5}
f_2(t,y,u_1,p)-f_2(t,x,u_2,p)\geq 0,
\\ \label{i.6}
f_2(t,x,u_1,-p)-f_2(t,y,u_2,-p)\geq 0
\end{gather}
for $t\in [0,T]$, $-l\leq y< x\leq l$, $-M\leq u_1<u_2\leq M$,
$p\in [q_0,q_1]$. For the Dirichlet boundary value problem we
additionally suppose that
\begin{equation}\label{i.7}
uf_2(t,x,u,p)\leq 0, \quad  \text{for} \quad  x\in [-l,-l+min
\{\tau_0, 2l \}] \bigcup [l-min \{\tau_0,2l \}, l],
\end{equation}
for $t\in [0,T]$, $|u|\leq M$ and $p\in [-q_1,q_0]\cup [q_0,q_1]$,
where $\tau_0$, $q_0$, $q_1$ are specified below.

Concerning the function $f_1$ we assume that
\begin{equation}\label{i.8}
|f_1(t,x,u,p)|\leq F_r(t,x,u,p,r)\psi(|p|)
\end{equation}
for $(t,x)\in \overline{Q}_T$, $|u|\leq M$ and arbitrary $(p,r)$,
where $\psi(\rho)\in {\bf{C}}^1(0,+\infty)$ is a nondecreasing
nonnegative function that satisfies the following condition: there
exist $q_0$ and $q_1$ such that $0<K\leq q_0<q_1<+\infty$ and
\begin{equation}\label{i.9}
\int^{q_1}_{q_0}\frac{\rho d\rho}{\psi(\rho)}\geq \mathop{\rm osc}(u)\equiv
\max u - \min u.
\end{equation}
Here $K$ is a Lipschitz constant of the initial function which
satisfies the assumption
\begin{equation}\label{i.10}
|u_0(x)-u_0(y)|\leq K|x-y|.
\end{equation}
Introduce $h(\tau)$ as a solution of the following problem
$$
h''+\psi(|h'|)=0, \quad h(0)=0, \quad h(\tau_0)=\mathop{\rm osc} (u).
$$
Represent the solution of $h''+\psi(|h'|)=0$ in parametrical form
(using the standard substitution $h'(\tau)=q(h)$,
$\frac{dq}{d\tau}=q\frac{dq}{dh}$):
$$
h(q)=\int^{q_1}_q \frac{\rho d\rho}{\psi(\rho)}, \quad
\tau(q)=\int^{q_1}_q \frac{d\rho}{\psi(\rho)}.
$$
The parameter $q$ varies in the interval $[q_0,q_1]$ and we select
$q_0$, $q_1$ such that $0<K\leq q_0<q_1<+\infty$,
$h(q_0)=\mathop{\rm osc}(u)$ (this is possible due to \eqref{i.9}).
Put $\tau_0\equiv \tau(q_0)$.

If conditions \eqref{i.8}-\eqref{i.10} as well as conditions
\eqref{i.5}, \eqref{i.6} are fulfilled, then the gradient of a
bounded solution of problem \eqref{1.1}, \eqref{1.2}, \eqref{1.5} is
bounded by a constant depending only on $K$, $\psi$, $\mathop{\rm osc}(u)$. In
the case of problem \eqref{1.1}, \eqref{1.3}, \eqref{1.5} we need
additional assumption on $q_0$ in terms of functions $\sigma_{i}$ (see
\cite[Lemma 2]{TT}). For problem \eqref{1.1}, \eqref{1.4},
\eqref{1.5} assumptions \eqref{i.5}-\eqref{i.10} guarantee the
gradient estimate of a bounded solution depending only on $K$,
$\psi$ and $\mathop{\rm osc}(u)$.

Note that if $f_1$ satisfies \eqref{i.3}, then for an arbitrary
Lipschitz continuous function $u_0(x)$ condition \eqref{i.9} is
automatically fulfilled for any $K$ (taking $q_1$ large enough and
using the divergence of the integral in \eqref{i.9} in this case).

Consider conditions \eqref{i.5}, \eqref{i.6}. These conditions
guarantee that the gradient of a bounded solution of equation
\eqref{1.1} cannot blow-up in the interior of $Q_T$ for any $T>0$ in
the case of problems \eqref{1.1}, \eqref{1.2}, \eqref{1.5} and
\eqref{1.1}, \eqref{1.3}, \eqref{1.5} (see also Remark \ref{rmk1}).
When $f_2$ is independent of $x$, one can easily see that
\eqref{i.5}, \eqref{i.6} mean that $f_2(t,u,p)$ is a nonincreasing
function with respect to $u$. Unfortunately if $f_2$ depends also on
$x$ and satisfies \eqref{i.5}, \eqref{i.6}, its behavior becomes
rather complicated.

The goal of this paper is to show that under some additional
assumptions the strict monotonicity of $f_2(t,x,u,p)$ in $u$ is
sufficient for nonexistence of the gradient blow-up of a bounded
solution. In order to motivate these additional assumptions we will
recall some facts from the theory of viscosity solutions. One can
easily see that in the case where $f_2$ is independent of $x$,
conditions \eqref{i.5}, \eqref{i.6} reminds us one of the main
assumptions (the properness, see \cite{CLI}) under which the notion
of viscosity solution is introduced. For example, if we assume in
\eqref{i.2} that $F_r$ is independent of $u$ and
$F(t,x,u,p,0)=f_2(t,u,p)$ satisfies \eqref{i.5}, \eqref{i.6}, then
\begin{equation}\label{in.1}
u_t-F_r(t,x,u_x,\lambda u_{xx})u_{xx}-f_2(t,u,u_x)=0,\quad
\lambda\in [0,1],
\end{equation}
is proper. Moreover in \cite{KK}, in particular, it was shown that
if $F_r=F_r(p,r)$ is locally strictly elliptic and
$F(t,x,u,p,0)=f_2(u,p)$ satisfies \eqref{i.5}, \eqref{i.6}, then
there exists a unique continuous viscosity solution to the Dirichlet
problem
\begin{equation}\label{in.2}
\begin{gathered}
u_t-F_r(u_x,\lambda u_{xx})u_{xx}-f_2(u,u_x)=0,\quad \lambda\in
[0,1], \\
u(t,-l)=u(t,l)=0, \quad u(0,x)=u_0(x),
\end{gathered}
\end{equation}
provided (\ref{in.2}) has a sub- and supersolution satisfying
initial-boundary data. Comparing this result with the results of
\cite{TT}, we conclude that a viscosity solution of the mentioned
above problem becomes classical, if additionally $f_2$ satisfies
\eqref{i.7} and the coefficients have sufficient smoothness. The
situation becomes more complicated, when $F_r$ and $f_2$ depends
also on $t$ and $x$. First of all we have to assume that the
elliptic operator is uniformly proper. It means that $f_2$ is
strictly decreasing in $u$
\begin{equation}\label{n.1}
f_2(t,x,u_1,p)-f_2(t,x,u_2,p)\geq \gamma_0(u_2-u_1)
\end{equation}
for $u_2\geq u_1$, $x\in[-l,l]$, $p\in {\bf{R}}$, for fixed $t\in
[0,T]$, where $\gamma_0$ is a positive constant. The second
assumption is a structure condition on the continuity of the
elliptic operator in $x$ (see \cite{CLI}). Assumptions that we use
in order to improve \eqref{i.5}, \eqref{i.6} were inspired by these
two assumptions under which the existence of a viscosity solution
can be proved.

We proceed now to the statement of main results of the paper. Assume
that $F(t,x,u,p,r)$ is defined for $(t,x)\in \overline{Q}_T$, $u\in
[-M,M]$ and arbitrary $(p,r)$ and is bounded on every compact set in
$\overline{Q}_T\times [-M,M]\times {\bf{R}}^2$. Suppose that
\begin{equation}\label{n.3}
|f_2(t,x,u,p)-f_2(t,y,u,p)|\leq K_1(t,x,y,u,p)|x-y|
\end{equation}
for $t\in[0,T]$, $x,y\in[-l,l]$, $0<x-y<\tau_0$, $|u|\leq M$, $p\in
[-q_1, -q_0]\cup [q_0, q_1]$, where $K_1\geq 0$,
\begin{equation}\label{n.4}
f_2(t,x,u_1,p)-f_2(t,x,u_2,p)\geq \gamma(t,x,u_1,u_2,p)(u_2-u_1)
\end{equation}
for $t\in[0,T]$, $x\in[-l,l]$, $|u_1|, |u_2|\leq M$, $u_2\geq u_1$,
$p\in [-q_1, -q_0]\cup [q_0, q_1]$, where $\gamma(t,x,u_1,u_2,p)\geq
\gamma_0>0$. Denote by ${\bf{V}}$ the following set
$$
{\bf{V}}=\{(t,x,y)\in \overline{Q}_T, 0<x-y<\tau_0, |u_1|, |u_2|\leq
M, u_2\geq u_1, p\in [-q_1, -q_0]\cup [q_0, q_1]\}.
$$
Assume that
\begin{equation}\label{n.7}
\max_{{\bf{V}}}\frac{K_1(t,x,y,u_1,p)}{\gamma(t,x,u_1,u_2,p)}\leq
C|p|^{\alpha},
\end{equation}
where $\alpha<1$ and $C$ is a positive constant.

Consider problem \eqref{1.1}, \eqref{1.2}, \eqref{1.5}.

\begin{theorem}\label{a}
Let $u(t,x)$ be a classical solution of problem \eqref{1.1},
\eqref{1.2}, \eqref{1.5}. Suppose that conditions \eqref{i.1},
\eqref{i.4}, \eqref{i.8}-\eqref{i.10},
\eqref{n.3}-\eqref{n.7} are fulfilled. Then in $\overline{Q}_T$
the inequality
$$
|u_x(t,x)|\leq C_1
$$
holds, where the constant $C_1$ depends on $\mathop{\rm osc}(u)$, $\psi$, $C$
and $\alpha$.
\end{theorem}

 From this theorem it follows that the gradient of a bounded
solution of \eqref{i.2} cannot blow-up in the interior of $Q_T$ for
any $T>0$ for problem \eqref{1.1}, \eqref{1.2}, \eqref{1.5}.
Analogous results we obtain for problem \eqref{1.1}, \eqref{1.3},
\eqref{1.5} (see Corollary \ref{b}). In the case of problem
\eqref{1.1}, \eqref{1.4}, \eqref{1.5} we also need assumption
\eqref{i.7} to obtain the nonexistence of the gradient blow-up on
the boundary as well as in the interior of $Q_T$ (see Corollary
\ref{c}). Note (see Remark \ref{rmk1}) that if $f_2(t,x,0,p)=0$,
then condition \eqref{i.7} is a simple consequence of the strict
monotonicity of $f_2$ in $u$ (see condition \eqref{n.4}).

Consider now the case when $f_2(t,x,u,p)=X(t,x)U(u)H(p)$ (special
case). Suppose that
\begin{equation}\label{3.1}
|f_2(t,x,u,p)-f_2(t,y,u,p)|\leq K_2|U(u)||H(p)||x-y|
\end{equation}
for $t\in[0,T]$, $x,y\in[-l,l]$, $0<x-y<\tau_0$, $|u|\leq M$, $p\in
[-q_1, -q_0]\cup [q_0, q_1]$, where $K_2$ is a Lipschitz constant of
$X(t,x)$ with respect to $x$,
\begin{equation}\label{3.2}
f_2(t,x,u_1,p)-f_2(t,x,u_2,p)=X(t,x)H(p)(U(u_1)-U(u_2))\geq
\end{equation}
$$
\gamma_1X(t,x)H(p)(u_2-u_1)\geq \gamma_0(u_2-u_1)
$$
for $t\in[0,T]$, $x\in[-l,l]$, $|u_1|, |u_2|\leq M$, $u_2>u_1$,
$p\in [-q_1, -q_0]\cup [q_0, q_1]$, where $\gamma_1,\gamma_0>0$ and
without loss of generality we assume that $U(u(t,x))$ is a strictly
decreasing function. Put $\gamma_2=\min_{t,x\in [0,T]\times
[-l,l]}|X(t,x)|>0$.

Consider problem \eqref{1.1}, \eqref{1.2}, \eqref{1.5}.

\begin{theorem}\label{d}.
Let $u(t,x)$ be a classical solution of problem \eqref{1.1},
\eqref{1.2}, \eqref{1.5}. Suppose that conditions \eqref{i.1},
\eqref{i.4}, \eqref{i.8} - \eqref{i.10}, \eqref{3.1},
\eqref{3.2} are fulfilled. Then in $\overline{Q}_T$ the inequality
$$
|u_x(t,x)|\leq C_4
$$
holds, where the constant $C_4$ depends on $\mathop{\rm osc}(u)$, $M$, $\psi$,
$K_2$, $\gamma_1$ and $\gamma_2$.
\end{theorem}

 From Theorem \ref{d} it follows that the gradient of a bounded
solution of \eqref{i.2} cannot blow-up in the interior of $Q_T$ for
any $T>0$ if $f_2=X(t,x)U(u)H(p)$ is Lipschitz continuous in $x$ and
strictly decreasing in $u$. Concerning the special case see also
Remark \ref{rmk5}.

Comparing Theorem \ref{a} with Theorem \ref{d} one can easily see
that in the case where $f_2$ is an arbitrary function of variables
$t$, $x$, $u$, $p$, besides the Lipschitz continuity in $x$ and
strict monotonicity in $u$, we need to impose some additional
structure conditions regarding the behavior of $f_2$ in $p$ (see
also Remark \ref{rmk3}) .

In Section 1 we obtain the a priori estimate of the gradient of a
bounded solution in the general case $f_2=f_2(t,x,u,p)$. In Section
2 we obtain the a priori estimate of the gradient of a bounded
solution in the case where $f_2=X(t,x)U(u)H(p)$.

We remark that based on these a priori estimates one can prove the
existence theorems for the initial-boundary value problems for
\eqref{1.1}, using the well-known fixed point theorem (see
\cite{LIB}). The proofs are exactly the same as in \cite{TT}.

\section{Gradient estimates in the general case}

In this section we  obtain global a priori estimates of the
gradient of classical solutions for boundary value problems for
 \eqref{1.1}, in the case where $f_2(t,x,u,p)$ is an
arbitrary bounded function of variables $t$, $x$, $u$, $p$. Recall
that a classical solution is a function belonging to
$C^{1,2}_{t,x}(Q_T)\cap C^{0,1}_{t,x}(\bar{Q}_T)$ in the case of
problem \eqref{1.1}, \eqref{1.2}, \eqref{1.5} or \eqref{1.1},
\eqref{1.3}, \eqref{1.5} and to
$C^{1,2}_{t,x}(Q_T)\cap C^{0}(\bar{Q}_T)$ for problem
\eqref{1.1}, \eqref{1.4}, \eqref{1.5}.
We use here  Kruzhkov's idea of introducing a new spatial
variable \cite{SNK,KRYJ} and the technique developed in
\cite{TT}.

\begin{proof}[Proof of Theorem \ref{a}]
Consider equation \eqref{1.1} in
the form \eqref{i.2} at two different points $(t,x)$ and $(t,y)$:
\begin{gather}\label{2.1}
u_t=F_r(t,x,u,u_x,\lambda u_{xx}) u_{xx} + F(t,x,u,u_x,0), \quad
\lambda\in [0,1], \quad u=u(t,x),
\\ \label{2.2}
u_t=F_r(t,y,u,u_y,\mu u_{yy}) u_{yy} + F(t,y,u,u_y,0), \quad
\mu\in [0,1], \quad u=u(t,y).
\end{gather}
Introduce the function $v(t,x,y)=u(t,x)-u(t,y)$.
In $\Omega=\{(t,x,y):0<t<T, 0<x-y, |x|<l, |y|<l \}$
the function $v(t,x,y)$ satisfies the equation
\begin{equation} \label{2.3}
\begin{aligned}
&-v_t+F_r(t,x,u(t,x),u_x(t,x),\lambda u_{xx}(t,x)) v_{xx} \\
&+ F_r(t,y,u(t,y),u_y(t,y),\mu u_{yy}(t,y)) v_{yy}\\
&= F(t,y,u(t,y),u_y(t,y),0)-F(t,x,u(t,x),u_x(t,x),0).
\end{aligned}
\end{equation}
Put
\[
F^{(x)}_r=F_r(t,x,u(t,x),u_x(t,x),\lambda u_{xx}(t,x)), \;
F^{(y)}_r= F_r(t,y,u(t,y),u_y(t,y),\mu u_{yy}(t,y)).
\]
Define the operator
$$
L(v)\equiv -v_t + F^{(x)}_r[v_{xx} + \psi(|v_x|)] +
F^{(y)}_r[v_{yy} + \psi(|v_{y}|)].
$$
 From (\ref{i.4}), \eqref{i.8} it follows that
\begin{equation}\label{2.4}
L(v)\geq f_2(t,y,u(t,y),u_y(t,y)) - f_2(t,x,u(t,x),u_x(t,x)).
\end{equation}
Let the function $h(\tau)$ be a solution of the ordinary
differential equation
\begin{equation}\label{2.5}
h''(\tau) + \psi(|h'(\tau)|)=0
\end{equation}
on the interval $[0,\tau_0]$ and satisfies conditions:
\begin{equation}\label{2.6}
h(0)=0, \quad h(\tau_0)=\mathop{\rm osc}(u), \quad h'>0 \quad
\text{for} \quad \tau\in [0,\tau_0].
\end{equation}
Represent the solution of \eqref{2.5}, \eqref{2.6} in parametrical
form
$$
h(q)=\int^{q_1}_q \frac{\rho d\rho}{\psi(\rho)}, \quad
\tau(q)=\int^{q_1}_q \frac{d\rho}{\psi(\rho)}.
$$
The parameter $q$ varies in the interval $[q_0,q_1]$, where
 $K^*< q_0<q_1<+\infty$, $K^*=\max\big\{K, C^{\frac{1}{1-\alpha}}\big\}$
and
\begin{equation}\label{2.61}
h(q_0)=\int^{q_1}_{q_0} \frac{\rho d\rho}{\psi(\rho)}=osc(u).
\end{equation}
Put
$$
\tau_0\equiv \tau(q_0)=\int^{q_1}_{q_0}\frac{d\rho}{\psi(\rho)}.
$$
Consider the function $w(t,x,y)=v(t,x,y)-h(x-y)$ in
$P=\{(t,x,y):0<t<T, 0<x-y<\tau_0, |x|<l,|y|<l \}$. Due to the fact
that $h(\tau)$ satisfies \eqref{2.5} we have $L(h(x-y))=0$. Hence,
using (\ref{2.4}) we obtain
\begin{align*}
\widetilde{L}(w)
&\equiv L(v)-L(h)\equiv -w_t + F^{(x)}_r[w_{xx} +
\alpha_1w_x] + F^{(y)}_r[w_{yy}+ \alpha_2w_{y}]\\
&\geq f_2(t,y,u(t,y),u_y(t,y)) - f_2(t,x,u(t,x),u_x(t,x)).
\end{align*}
Where $|\alpha_i|< +\infty$, $i=1,2$, by virtue of the mean value
theorem and of the fact that $\psi$ is a smooth function and $u$
is a classical solution of \eqref{1.1}, \eqref{1.2}, \eqref{1.5}.
Let $\tilde{w}=we^{-t}$, then
\begin{equation}\label{31.1}
\begin{aligned}
\widetilde{L}_1(\tilde{w})
&\equiv -\tilde{w}_t + F^{(x)}_r[\tilde{w}_{xx} + \alpha_1\tilde{w}_x] +
F^{(y)}_r[\tilde{w}_{yy}+ \alpha_2\tilde{w}_y]-\tilde{w}\\
&\geq e^{-t}[f_2(t,y,u(t,y),u_y(t,y)) - f_2(t,x,u(t,x),u_x(t,x))].
\end{aligned}
\end{equation}
Denote by $\Gamma$ the parabolic boundary of $P$ (i.e.
$\Gamma=\partial{P} \setminus \{(t,x,y): t=T, 0<x-y<\tau_0, |x|<l,
|y|<l \}$). Suppose that the function $\tilde{w}$ attains its
positive maximum at some point $(t_1,x_1,y_1) \in
\overline{P}\setminus \Gamma$. Obviously it should be
$\widetilde{L}_1(\tilde{w})\big|_{(t_1,x_1,y_1)}<0$. On the other
hand, at this point we have
$$
-\tilde{w}<0, \quad \tilde{w}_x=\tilde{w}_y=0, \quad
\tilde{w}_{xx}\leq 0,\quad \tilde{w}_{yy}\leq 0, \quad
-\tilde{w}_t\leq 0;
$$
i.e.,
\begin{gather*}
\tilde{w}(t_1,x_1,y_1)=e^{-t}[u(t_1,x_1) - u(t_1,y_1) -
h(x_1-y_1)]>0,
\\
\tilde{w}_x(t_1,x_1,y_1)=e^{-t}[u_{x}(t_1,x_1) - h'(x_1-y_1)]=0,
\\
\tilde{w}_y(t_1,x_1,y_1)=e^{-t}[-u_{y}(t_1,y_1) + h'(x_1-y_1)]=0
\end{gather*}
and as a consequence
\begin{equation}\label{31.2}
u(t_1,x_1)>u(t_1,y_1), \quad
u_{x}(t_1,x_1)=u_{y}(t_1,y_1)=h'(x_1-y_1)>0.
\end{equation}
Represent the right-hand side of inequality \eqref{31.1} in the
following way
\begin{equation}\label{n1.1}
\begin{aligned}
&e^{-t}[f_2(t,y,u(t,y),u_y(t,y)) - f_2(t,x,u(t,x),u_x(t,x))]\\
&=e^{-t}[f_2(t,y,u(t,y),u_y(t,y)) -f_2(t,x,u(t,y),u_y(t,y))\\
&\quad + f_2(t,x,u(t,y),u_y(t,y)) - f_2(t,x,u(t,x),u_x(t,x))],
\end{aligned}
\end{equation}
where we subtract and add the term $f_2(t,x,u(t,y),u_y(t,y))$. So
at the maximum point $(t_1,x_1,y_1)$, using \eqref{n.3},
\eqref{n.4}, (\ref{31.2}), we obtain
\begin{equation}\label{n1.2}
\begin{aligned}
&\widetilde{L}_1(\tilde{w})\\
&\geq e^{-t_1}[f_2(t_1,y_1,u(t_1,y_1),u_y(t_1,y_1)) -
f_2(t_1,x_1,u(t_1,y_1),u_y(t_1,y_1))\\
&\quad + f_2(t_1,x_1,u(t_1,y_1),u_y(t_1,y_1)) -
f_2(t_1,x_1,u(t_1,x_1),u_x(t_1,x_1))]\\
&\geq e^{-t_1}\Big[-K_1\Big(t_1,x_1,y_1,u(t_1,y_1),h'(x_1-y_1)\Big)(x_1-y_1)\\
&\quad + \gamma\Big(t_1,x_1,y_1,u(t_1,x_1),u(t_1,y_1),h'(x_1-y_1)\Big)
(u(t_1,x_1)-u(t_1,y_1))\Big].
\end{aligned}
\end{equation}
Consider now the difference $u(t_1,x_1)-u(t_1,y_1)$. Due to the
fact that
$$
\tilde{w}(t_1,x_1,y_1)=e^{-t}[u(t_1,x_1) - u(t_1,y_1) -
h(x_1-y_1)]>0,
$$
we have
$$
u(t_1,x_1) - u(t_1,y_1)>
h(x_1-y_1)=h(x_1-y_1)-h(0)=h'(\xi)(x_1-y_1)
$$
for some $\xi\in [0, \tau_0]$. Thus one can rewrite the inequality
(\ref{n1.2}) in the following way
\begin{equation}\label{n1.3}
\begin{aligned}
\widetilde{L}_1(\tilde{w})
&\geq e^{-t_1}\Big[-K_1\Big(t_1,x_1,y_1,u(t_1,y_1),h'(x_1-y_1)\Big)\\
&\quad + \gamma\Big(t_1,x_1,y_1,u(t_1,x_1),u(t_1,y_1),h'(x_1-y_1)\Big)
h'(\xi)\Big](x_1-y_1).
\end{aligned}
\end{equation}
Using now \eqref{n.7} we conclude that
\begin{equation}\label{n1.4}
\widetilde{L}_1(\tilde{w})\geq
e^{-t_1}\Big[-Ch'^{\alpha}(x_1-y_1)+
h'(\xi)\Big]\gamma_0(x_1-y_1).
\end{equation}
To obtain the contradiction with
$\widetilde{L}_1(\tilde{w})\big|_{(t_1,x_1,y_1)}<0$ we have to
show that (recall that $x_1>y_1$)
$$
-Ch'^{\alpha}(x_1-y_1)+ h'(\xi)\geq 0.
$$
Using the fact that $q_0\leq h'\leq q_1$, we arrive to the
inequality
$$
-Cq^{\alpha}_1+q_0\geq 0.
$$
Thus if
\begin{equation}\label{n1.5}
q_0\geq Cq_1^{\alpha},
\end{equation}
then $\widetilde{L}_1(\tilde{w})\big|_{(t_1,x_1,y_1)}\geq 0$.
Obviously, when $\alpha<1$, there exist $q_0$ and $q_1>q_0$ such
that inequality \eqref{n1.5} takes place. Solving the system
$$
q_0\geq Cq_1^{\alpha}, \quad q_1>q_0,
$$
one can easily obtain that for $q_0>C^{\frac{1}{1-\alpha}}$ inequality
\eqref{n1.5} takes place for
$q_0<q_1\leq\left(\frac{1}{C}\right)^{\frac{1}{\alpha}}q_0^{\frac{1}{\alpha}}$.
Consequently it follows that $\tilde{w}$ cannot attain its positive
maximum in $\bar{P}\setminus \Gamma$. Note that if $\alpha\leq 0$
then the validity of \eqref{n1.5} does not depend on $q_1$.

Now let us show that $\tilde{w}\big|_{\Gamma}\leq 0$. Consider two
possible cases: $\tau_0<2l$ and $\tau_0\geq 2l$. First let
$\tau_0<2l$. For $t=0$:
$$
\tilde{w}(0,x,y)=e^{-t}(u_0(x)-u_0(y) - h(x-y))\leq e^{-t}(K(x-y) -
h'(\tau^*)(x-y))\leq 0,
$$
where $\tau^*\in [0, \tau_0]$, due to the fact that $h'\geq q_0\geq
K$. Obviously $\tilde{w}(t,x,y)\big|_{x=y}=0$ and when $x-y=\tau_0$
we have $\tilde{w}=e^{-t}(u(t,x) - u(t,y) - h(\tau_0))\leq 0$ due to
\eqref{2.6}. Denote by $Q_1=\{(t,x):0<t\leq T, -l<x<-l+\tau_0, y=-l
\}$, $Q_2=\{(t,y):0<t\leq T, l-\tau_0 <y<l, x=l \}$. Estimate the
normal derivative of $\tilde{w}$ on $Q_1$ and $Q_2$ using boundary
conditions \eqref{1.2} and the fact that $h'\geq q_0>0$
\begin{gather*}
-\tilde{w}_y(t,x,-l)=e^{-t}(u_{y}(t,-l) -
h'(x+l))=-e^{-t}h'(x+l)<0,
\\
\tilde{w}_x(t,l,y)= e^{-t}(u_{x}(t,l) - h'(l-y))=-e^{-t}h'(l-y)<0.
\end{gather*}
Thus the function $\tilde{w}(t,x,y)$ cannot attain its positive
maximum neither on $Q_1$ nor on $Q_2$ since $-\partial/\partial y$
and $\partial/\partial x$ are here outward normal derivatives with
respect to $P$. Consequently, $\tilde{w}\big|_{\Gamma}\leq 0$ and
hence $\tilde{w}(t,x,y)\leq 0$ in $\overline{P}$.

The case when $\tau_0\geq 2l$ can be treated similarly. The only
difference is the absence of the boundary $x-y=\tau_0$. We put
$\widetilde{Q}_1=\{(t,x):0<t\leq T, -l<x\leq l, y=-l \}$,
$\widetilde{Q}_2=\{(t,y):0<t\leq T, -l<y<l, x=l \}$ (note that the
line $x=l, y=-l$ belongs to $\widetilde{Q}_1)$. Consequently,
$\tilde{w}\big|_{\Gamma}\leq 0$ and hence $\tilde{w}(t,x,y)\leq 0$
in $\overline{P}$. It means that
\begin{equation}\label{41.3}
u(t,x) - u(t,y)\leq h(x-y) \quad {\rm in} \quad \overline{P}.
\end{equation}
Treating similarly the function $\tilde{v}(t,x,y)=u(t,y) - u(t,x)$
one can easily see that for
$\tilde{w}_1(t,x,y)=e^{-t}(\tilde{v}(t,x,y) -h(x-y))$ we have
$$
\widetilde{L}_1(\tilde{w}_1)\geq e^{-t}[f_2(t,x,u(t,x),u_x(t,x)) -
f_2(t,y,u(t,y),u_y(t,y))] \quad \text{in} \quad P.
$$
Suppose that $\tilde{w}_1$ attains its positive maximum at
$(\tilde{t}_1,\tilde{x}_1,\tilde{y}_1)\in \overline{P}\setminus
\Gamma$. Consequently it should be
$\widetilde{L}_1(\tilde{w}_1)\big|_{(\tilde{t}_1,\tilde{x}_1,\tilde{y}_1)}<0$.
On the other hand, we have
$$
u(\tilde{t}_1,\tilde{y}_1)>u(\tilde{t}_1,\tilde{x}_1), \quad
u_x(\tilde{t}_1,\tilde{x}_1)=u_{y}(\tilde{t}_1,\tilde{y}_1)=
-h'(\tilde{x}_1-\tilde{y}_1)<0.
$$
Using inequalities \eqref{n.3} - \eqref{n.7} we obtain in the same
way that $\widetilde{L}_1(\tilde{w}_1)\geq 0$. From this
contradiction it follows that $\tilde{w}_1$ cannot attain its
positive maximum in $\overline{P}\setminus \Gamma$.

Consider $\tilde{w}_1$ on $\Gamma$. One can easily see that all
considerations concerning the estimate of the function $\tilde{w}$
on the boundary $\Gamma$ can be done without any changes in estimate
of $\tilde{w}_1$. Thus we have that
\begin{equation}\label{5.1}
u(t,y) - u(t,x)\leq h(x-y) \quad {\rm in} \quad \overline{P}.
\end{equation}
Combining (\ref{5.1}) with (\ref{41.3}) we get
$$
|u(t,x) - u(t,y)|\leq h(x-y) \quad \text{in } \overline{P}.
$$
In view of the symmetry of the variables $x$, $y$ in the same
manner we examine the case $y>x$. As a result we have that for
$$
0\leq t\leq T, \quad |x|\leq l, \quad |y|\leq l, \quad 0<|x-y|\leq
\tau_0
$$
the inequality
$$
\big|\frac{u(t,x) - u(t,y)}{x-y}\big|
\leq \frac{h(|x-y|) -h(0)}{|x-y|}
$$
holds and as a consequence we have
$$
|u_x(t,x)|\leq h'(0)=q_1=C_1.
$$
Theorem \ref{a} is proved.
\end{proof}

Let us pass to problem \eqref{1.1}, \eqref{1.3}, \eqref{1.5}.

\begin{corollary}\label{b}
Let $u(t,x)$ be a classical solution of \eqref{1.1},
$\eqref{1.3}$, $\eqref{1.5}$ and all conditions of Theorem
$\ref{a}$ are fulfilled. Then in $\overline{Q}_T$ the inequality
$$
|u_x(t,x)|\leq C_2
$$
holds, where the constant $C_2$ depends only on $\mathop{\rm osc}(u)$, $N_1$,
$N_2$, $\psi$, $C$, $\alpha$, where $N_i=\sup|\sigma_i|$ (the
supremum is taken over the set $[0,T]\times [-M,M]$).
\end{corollary}

\begin{proof}
The proof of this corollary differs from
the proof of Theorem \ref{a} only in the selection of $q_0$ and in
analyzing of the behavior of $\tilde {w}(t,x,y)$ on bounds $Q_1$
($\widetilde{Q}_1$) and $Q_2$ ($\widetilde{Q}_2$). We select the
quantity $q_0$ so that
\begin{equation}\label{6.1}
q_0>max \left \{C^{\frac{1}{1-\alpha}},K, N_1, N_2 \right \}.
\end{equation}
and follow the proof of \cite[Lemma 2, and Corollary  1.2]{TT}.
Corollary \ref{b} is proved.
\end{proof}

Consider now problem \eqref{1.1}, \eqref{1.4}, \eqref{1.5}.

\begin{corollary}\label{c}.
Let $u(t,x)$ be a classical solution of \eqref{1.1},
\eqref{1.4}, \eqref{1.5} and all conditions of Theorem
\ref{a} are fulfilled. Suppose in addition that condition
\eqref{i.7} is fulfilled and $u_0(\pm l)=0$. Then in
$\overline{Q}_T$ the following inequality
$$
|u_x(t,x)|\leq C_3
$$
holds, where the constant $C_3$ depends only on $\mathop{\rm osc}(u)$ and
$\psi$, $C$, $\alpha$.
\end{corollary}

\begin{proof}
 The proof of this corollary  differs from
the proof of Theorem $\ref{a}$ only in analyzing  of the
behavior of $w(t,x,y)$ on $Q_1$ ($\tilde{Q}_1)$ and $Q_2$
($\tilde{Q}_2)$ (see the proof of \cite[Lemma 3 and Corollary 1.3]{TT}).
 Corollary \ref{c} is proved.
\end{proof}

\begin{remark} \label{rmk1} \rm
 One can easily see that from Theorem \ref{a} and
Corollary \ref{b} it immediately follows that conditions
\eqref{n.3}-\eqref{n.7} are sufficient for the nonexistence of the
gradient blow-up of a bounded solution in the interior of $Q_T$ for
problems \eqref{1.1}, \eqref{1.2}, \eqref{1.5} and \eqref{1.1},
\eqref{1.3}, \eqref{1.5}. Concerning problem \eqref{1.1},
\eqref{1.4}, \eqref{1.5}, one has to impose condition \eqref{i.7}
supplementary to \eqref{n.3}-\eqref{n.7} in order to obtain the
nonexistence of the gradient blow-up of a bounded solution in
$\overline{Q}_T$. Obviously, if we suppose that $f_2(t,x,0,p)=0$,
then condition \eqref{i.7} is a simple consequence of the strict
monotonicity of $f_2$ in $u$ (see condition \eqref{n.4}) Thus if
$f_2(t,x,0,p)=0$, then the nonexistence of the gradient blow-up in
$\overline{Q}_T$ for problem \eqref{1.1}, \eqref{1.4}, \eqref{1.5}
follows from \eqref{n.3}-\eqref{n.7} and we do not need condition
\eqref{i.7}.
\end{remark}

\begin{remark} \label{rmk2} \rm
Note that if $f_1$ satisfies Bernstein-Nagumo
condition \eqref{i.3} then for every $q_0$ there always exists
$q_1>q_0$ such that \eqref{i.9} takes place. Thus if $\alpha\leq 0$
in \eqref{n.7}, then inequality \eqref{n1.5} does not depend on
$q_1$ and we can always construct $h(\tau)$ that satisfies
\eqref{2.6} (due to the divergence of the integral in \eqref{i.9} in
this case).
\end{remark}

\begin{remark} \label{rmk3} \rm
Put $f_1=0$. In this case $\psi(p)=0$ and
$h''=0$. From \eqref{2.6} it follows that $h=\frac{osc\,
u}{\tau_0}\tau$, $h'=\frac{osc\, u}{\tau_0}$. Thus we have that
$q_0=q_1=\frac{osc\, u}{\tau_0}$. Consequently \eqref{n1.5} takes
the form
\begin{equation}\label{r3.1}
q_0\geq Cq_0^{\alpha}
\end{equation}
that is fulfilled for  $q_0\geq C^{\frac{1}{1-\alpha}}$ and
$\alpha<1$. Obviously, when $\alpha=1$, then in order to obtain the
gradient a priori estimate one has to suppose that $C\leq 1$. One
can easily check that even in the case where $f_1=0$ \eqref{r3.1}
holds for $q_0\leq C^{\frac{1}{1-\alpha}}$ if $\alpha>1$. Thus we
can prove Theorem \ref{a} with $\alpha>1$ only for $K\leq
C^{\frac{1}{1-\alpha}}$. Note that condition \eqref{n.7} can be
generalized in the following way
$$
\max_{{\bf{V}}}\frac{K_1(t,x,y,u_1,p)}{\gamma(t,x,u_1,u_2,p)}\leq
\Psi(|p|),
$$
where $\Psi(\rho)$ is a nondecreasing positive function. As a
consequence condition \eqref{n1.5} appears as
$$
q_0\geq \Psi(q_1).
$$
\end{remark}

\begin{remark} \label{rmk4} \rm
Let us give some simple examples of functions
that satisfy \eqref{n.3}-\eqref{n.7} and at the same time do not
satisfy \eqref{i.5}, \eqref{i.6}. Easy calculations show that, for
example, the functions
$$
f_2(t,x,u,p)=x|p|^{\mu}-u|p|^{\nu}, \quad \forall \quad \mu,\nu
\quad \hbox{such that } \mu-\nu\leq 1,
$$
$$
f_2(t,x,u,p)=-ue^{(x+c)p^{\alpha}}, \quad \alpha<1, \; c>l,
$$
where $\alpha$ is such that $p^{\alpha}$ is defined, satisfy
\eqref{n.3}-\eqref{n.7} and do not satisfy \eqref{i.5}, \eqref{i.6}.
Moreover, in the case when $f_2=-ue^{(x+c)p^{\alpha}}$, $\alpha<1$,
$c>l$, problem \eqref{1.4}, \eqref{1.5} as well as problems
\eqref{1.2}, \eqref{1.5} and \eqref{1.3}, \eqref{1.5}, for example,
for the equation
\begin{equation}\label{sup.1}
u_t=a(t,x,u,u_x)u_{xx}+f_2, \quad a>0,
\end{equation}
have a global classical solution for any Lipschitz continuous
initial data. When $f_2(t,x,u,p)=x|p|^{\mu}-u|p|^{\nu}$, $\mu-\nu\leq
1$, problems \eqref{sup.1}, \eqref{1.2}, \eqref{1.5} and
\eqref{sup.1}, \eqref{1.3}, \eqref{1.5} have a global classical
solution for any Lipschitz continuous initial data.
\end{remark}

\section{Gradient estimates in the special case}

In this section we obtain global a priori estimates of the
gradient of classical solutions for boundary-value problems for
equation \eqref{1.1} where $f_2=X(t,x)U(u)H(p)$. One can easily see
that in this case conditions \eqref{n.3}, \eqref{n.4} take the
 form
$$
|f_2(t,x,u,p)-f_2(t,y,u,p)|\leq K_2|U(u)||H(p)||x-y|
$$
for $t\in[0,T]$, $x,y\in[-l,l]$, $0<x-y<\tau_0$, $|u|\leq M$, $p\in
[-q_1, -q_0]\cup [q_0, q_1]$, where $K_2$ is the Lipschitz constant
of $X(t,x)$ with respect to $x$,
\begin{align*}
f_2(t,x,u_1,p)-f_2(t,x,u_2,p)
&=X(t,x)H(p)(U(u_1)-U(u_2))\\
&\geq \gamma_1X(t,x)H(p)(u_2-u_1)\geq \gamma_0(u_2-u_1)
\end{align*}
for $t\in[0,T]$, $x\in[-l,l]$, $|u_1|, |u_2|\leq M$, $u_2>u_1$,
$p\in [-q_1, -q_0]\cup [q_0, q_1]$, where $\gamma_1,\gamma_0>0$.
Recall that without loss of generality we assume that $U(u(t,x))$ is
a strictly decreasing function. The last assumption means that
$X(t,x)H(p)>0$. Put $\gamma_2=\min_{t,x\in [0,T]\times
[-l,l]}|X(t,x)|>0$. Obviously if $f_2(t,x,u,p)=X(t,x)U(u)H(p)$, then
condition \eqref{n.7} is fulfilled with
$C=\frac{K_2|U(-M)|}{\gamma_1\gamma_2}$, $\alpha=0$ and as a consequence
can be dropped.

\begin{proof}[Proof of Theorem \ref{d}]
The proof differs from the proof of the previous theorem only in the
choice of the quantity $q_0$. In the general case $q_0$ depends on $q_1$
and $\alpha$ (see \eqref{n1.5}). We will show now that if
$f_2(t,x,u,p)=X(t,x)U(u)H(p)$, then $q_0$ is independent of $q_1$
and $\alpha$. Following the proof of Theorem \ref{a} we arrive to
(\ref{n1.3}) that appears in the  form
\begin{equation}\label{3.30}
\begin{aligned}
\widetilde{L}_1(\tilde{w})
&\geq e^{-t_1}[-K_2|U(u(t_1,y_1))||H(h'(x_1-y_1))|(x_1-y_1)\\
&\quad +\gamma_1 X(t_1,x_1)H(h'(x_1-y_1))(u(t_1,x_1))-u(t_1,y_1))].
\end{aligned}
\end{equation}
Due to \eqref{3.2} we have that $\gamma_1 X(t,x)H(p)>0$ and
consequently
\begin{equation}\label{3.300}
\gamma_1 X(t,x)H(p)=\gamma_1 |X(t,x)||H(p)|.
\end{equation}
Thus from (\ref{3.30}), (\ref{3.300}) we obtain that (recall that
$U(u(t,x))$ is a strictly decreasing function and $|u|\leq M$)
\begin{equation}\label{3.31}
\begin{aligned}
\widetilde{L}_1(\tilde{w})
&\geq e^{-t_1}|H(h'(x_1-y_1))|[-K_2|U(-M)|(x_1-y_1)\\
&\quad + \gamma_1|X(t_1,x_1)|(u(t_1,x_1)-u(t_1,y_1))].
\end{aligned}
\end{equation}
Due to the fact that $u(t_1,x_1))-u(t_1,y_1)\geq
h'(\xi)(x_1-y_1)$, from (\ref{3.31}) it follows (recall that
$h'(\xi)\geq q_0$)
\begin{equation}\label{3.4}
\begin{aligned}
\widetilde{L}_1(\tilde{w})
&\geq e^{-t_1}|H(h'(x_1-y_1))|(x_1-y_1)[-K_2|U(-M)|+\gamma_1
|X(t_1,x_1)|h'(\xi)]\\
&\geq e^{-t_1}|H(h'(x_1-y_1))|(x_1-y_1)[-K_2|U(-M)|+\gamma_1
|X(t_1,x_1)|q_0].
\end{aligned}
\end{equation}
To obtain the contradiction with
$\widetilde{L}_1(\tilde{w})\big|_{(t_1,x_1,y_1)}<0$ we have to show
that
\begin{equation}\label{3.5}
-K_2|U(-M)|+\gamma_1 |X(t_1,x_1)|q_0\geq 0.
\end{equation}
Obviously this is the case when
\begin{equation}\label{3.6}
q_0\geq \frac{K_2|U(-M)|}{\gamma_1\gamma_2}.
\end{equation}
 From this contradiction we conclude that $\tilde{w}$ cannot attain
its positive maximum in $\bar{P}\setminus \Gamma$. Similarly one can
prove that $\tilde{w}_1=e^{-t}(u(t,y)-u(t,x)-h(x-y))$ cannot attain
its positive maximum in $\bar{P}\setminus \Gamma$. Further without
any changes we follow the proof of Theorem \ref{a}. Note here that
in (\ref{2.61}) one has to suppose that $q_0\geq
\max\{K,\frac{K_2|U(-M)|}{\gamma_1\gamma_2}\}$. Theorem
\ref{d} is proved.
\end{proof}

\begin{remark} \label{rmk5} \rm
 In the case of problems \eqref{1.1}, \eqref{1.3},
\eqref{1.5} and \eqref{1.1}, \eqref{1.4}, \eqref{1.5} one can easily
formulate results that are similar to those of Corollaries \ref{b}
and \ref{c}. The proofs of these results is an easy compilation of
the proofs of Theorems \ref{a}, \ref{d}, Corollary \ref{b} and of
Theorems \ref{a}, \ref{d}, Corollary \ref{c} respectively.
\end{remark}

\begin{remark} \label{rmk6} \rm
 From the mentioned above it follows that the strict
monotonicity of $f_2(t,x,u,p)=X(t,x)U(u)H(p)$ in $u$, coupled with
the Lipschitz continuity in $x$, guarantee the nonexistence of the
gradient blow-up of a bounded solution in the interior of $Q_T$ for
problems \eqref{1.1}, \eqref{1.2}, \eqref{1.5} and \eqref{1.1},
\eqref{1.3}, \eqref{1.5}. If additionally $f_2(t,x,0,p)=0$ then the
strict monotonicity of $f_2(t,x,u,p)$ in $u$, coupled with the
Lipschitz continuity in $x$, guarantee the nonexistence of the
gradient blow-up of a bounded solution in $\overline{Q}_T$ for
problem \eqref{1.1}, \eqref{1.4}, \eqref{1.5}. Concerning problem
\eqref{1.1}, \eqref{1.4}, \eqref{1.5} in the case where
$f_2(t,x,0,p)\not= 0$, one has to impose condition \eqref{i.7}
supplementary to \eqref{3.1}, \eqref{3.2} in order to obtain the
nonexistence of the gradient blow-up of a bounded solution in
$\overline{Q}_T$.
\end{remark}

\begin{remark} \label{rmk7} \rm
 Let us give some simple examples of functions
that satisfy \eqref{3.1}, \eqref{3.2} and at the same time do not
satisfy \eqref{i.5}, \eqref{i.6}. Easy calculations show that, for
example, the functions
$$
f_2(t,x,u,p)=g(x)up^{2m}, \quad f_2(t,x,u,p)=g(x)u^3|p|^{\nu},
\quad f_2(t,x,u,p)=g(x)u e^p,
$$
where $g(x)<0$ is an arbitrary Lipschitz continuous function, $m>1$
is an integer number, $\nu>2$ is a real number, satisfy \eqref{3.1},
\eqref{3.2} and do not satisfy \eqref{i.5}, \eqref{i.6}. Moreover,
in the case where $f_2$ has one of the above representations,
problems \eqref{sup.1}, \eqref{1.2}, \eqref{1.5}, \eqref{sup.1},
\eqref{1.3}, \eqref{1.5} and \eqref{sup.1}, \eqref{1.4}, \eqref{1.5}
have a global classical solution for any Lipschitz continuous
initial data.
\end{remark}


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\end{document}