N+1.
\end{equation}
Furthermore, if $u_0\ge 0$ and $f\ge 0$ then
we construct a solution such that $u\ge 0$.
\end{theo}
In the preceding theorem, we can be more precise about the Maximum
Principe (or M.P. in short). Indeed, if $g$ is nondecreasing with
respect to $u$, then, if $u_0 \ge v_0$ a.e. and $f \ge g$ a.e., we
can construct corresponding solutions, $u$ and $v$, satisfying $u
\ge v$ a.e. (where $v$ is the constructed solution of
(\ref{eq2})-(\ref{ic}) with $v_0$ and $g$ instead of $u_0$ and
$f$). If one has uniqueness of the solution of
(\ref{eq2})-(\ref{ic}) we then obtain the so called Maximum
Principle. See more precise results in the final remark of Section
\ref{uniq}.
\medskip
\noindent {\sc Remarks on the conditions.} The lower bound on $p$,
$p>p_1$, is not essential. It is due to the fact
that we do not want to get out of the
classical weak formulation where the gradient is a locally integrable
function. Exponents $1< p \leq p_1$ can be handled by using a proper
definition of gradient (see \cite {B6}) but we will not include the
calculations here in order to avoid complicating too much the presentation.
Note that the lower limit in the elliptic theory is $p_0 1$ for $p > p_1$.
\smallskip
\noindent $\bullet$ For $p>N $ the point $B$ is not valid because it
violates the conditions $q s+1.
\end{equation}
\end{theo}
The new condition $s(p-1)>1$ is necessary for small values of $p$
in order for the admissible set of exponents to be non-empty.
Another extremal point $C$ is now given by the coordinates
$q=r=ps/(s+1)$.
Taking into account the common point $A$ and comparing the slopes of the
critical lines implies that an improvement of the admissibility region
takes place if $(N(p-2)+p)/N
\frac{(\alpha-1)(p-q)r}{(q^\star-1)q(p-r)}.
\end{equation}
In order to fulfill this inequality we may choose a small value of $\alpha$,
always larger than $q/(p-q)$, by taking $m>0$ very small. Therefore, we
are reduced to check the limit case $\alpha-1=(2q-p)/(p-q)$ which gives
$$
\frac{q^\star-1}{q^\star}>\frac{2q-p}{q(p-r)}.
$$
Working out this formula gives the relation
\begin{equation}
\frac{(p-2)N+p}r+\frac Nq> N+1.
\end{equation}
This relation is complemented by the restrictions on $q$: $1\le q 0$
$$
\int_{B_\rho} | Du|^q\,dx\le
\Big(\int_{B_\rho} \frac{| Du|^p}{(1+|
u|)^{m+1}}\,dx\Big)^{q/p}
\Big(\int_{B_\rho} (1+| u|)^{\alpha}dx\Big)^{(p-q)/p},
$$
where $\alpha=(m+1)q/(p-q)$ as before. Rising to the power $r/q$ with $r 0$ so that $1< \alpha 0$ the following sequences are bounded
uniformly in $n>2\rho$:
$$
\begin{array}{c}
\{u_n\} \quad\mbox{in } L^r(0,T; W^{1,q}(B_\rho(0))), \\[3pt]
\{g(x,t,u_n)\} \quad \mbox{in } L^1(B_\rho(0)\times(0,T)), \\[3pt]
\{u'_{n}\} \quad \mbox{in } L^1(0,T; W^{-1,\delta}(B_\rho(0)))+
L^1(0,T; L^{1}(B_\rho(0))),
\end{array}
$$
for some $\delta >1$. Moreover, since one obtains that the sequence
$\{u'_n\}$ is bounded in
$L^1(0,$ $T;$ $W^{-1,1}$ $(B_\rho(0)))$, using compactness arguments (see
\cite{Si}) it is easy to see
that the sequence $\{u_n\}$ is relatively compact in
$L^1(Q_\rho)$. By a diagonal
process we may select a subsequence, also denoted by $\{u_n\}$, such that
$$
u_n\to u \quad \mbox{a.e. and in } \ L^1(0,T; L^1_{{\rm loc}}(\mathbb{R}^N)),
$$
and also $u_n\to u$ weakly in $L^r(0,T; W^{1,q}_{{\rm loc}}(\mathbb{R}^N))$ for $q,r$ as
in Theorem \ref{theo2}
and Corollary \ref{cor4}.
We want to pass to the limit in the equation in order to get
a solution of the original problem.
We need to prove first the convergence of the sequence
$\{g(x,t,u_n)\}$, and also the $a.e.$ convergence (up to a subsequence)
of the gradients of
$\{u_n\}$, which will imply the convergence of $\{Du_n\}$ to $Du$ in
$L^r(0,T; L^{q}_{{\rm loc}}(B_R))$, for any $R>0$.
Let us prove the result about $g(x,t,u_n)$, which is based on an
argument of local equi-integrability. We resume the notations and
calculations of Section \ref{sect-local} with slight variations. We
consider the function $\phi$ defined in (\ref{phi}) and we displace it by
an amount $t>0$ to get
$$
\phi^t(s)=\left\{ \begin{array}{ll} \phi(s-t),\quad & s\geq t\\
0, & |s| N$ implies by Morrey's
inequality, cf. \cite[Theorem 4.5.3.3]{EG}, that we have an estimate of
the form
\begin{equation}\label{sobol}
\|u\|_{L^\infty(B_{\rho})} \le C + C\Big(\int_{B_\rho}
| Du|^q\,dx\Big)^{1/q},
\end{equation}
where $C$ depends only on $\rho, q$ and the $L^1$ norm of $u$ in $B_\rho$,
which is uniformly bounded independently of $R$ thanks to the previous
estimate on $u$ in $L^\infty(0,T; L^1_{{\rm loc}}(\mathbb{R}^N))$. Continuing as before, we
can write for $r~~0$
since $s>p-1$ by assumption. Using Young's inequality we get
\begin{equation}
C_1 (1+ | u| )^{(m+1)(p-1)}\theta^{\gamma-p} \le
\frac 12 | u|^{s-1} u \phi(u)\theta^\gamma +
C_2 (1 + \theta^{\gamma-\frac{ps}{s-(m+1)(p-1)}}).
\label{coucouclock}
\end{equation}
In view of the last exponent we choose $\gamma> ps/(s-(m+1)(p-1))$ and then
\begin{eqnarray}\label{3est}
\lefteqn{\int_{B_R} \psi(u(x,T))\theta(x)^\gamma\,dx +
\frac 12\int_0^T \int_{B_R}
| Du|^p\phi'(u)\theta^\gamma\,dx\,dt }\nonumber
\\
\lefteqn{ +\frac 12 \int_0^T \int_{B_R}
| u|^{s-1} u \phi(u)\theta^\gamma\,dx\,dt } \\
&\le&
\int_{B_{2\rho}} \psi(u_0(x))\,dx +
C_3\int_0^T \int_{B_{2\rho}} | f|\,dx\,dt + C_3\,T|
B_{2\rho}|, \nonumber
\end{eqnarray}
where $| B_{2\rho}|$ denotes the volume of $ B_{2\rho}$ in $\mathbb{R}^N$.
In view of our assumptions on the data we conclude that
\begin{equation}\label{C4}
\int_0^T \int_{B_\rho}
\frac{| Du|^p}{(1+| u|)^{m+1}}\,dx\,dt \le C_4,
\quad \int_0^T \int_{B_\rho}
| u|^{s}\,dx\,dt\le C_4,
\end{equation}
where $C_4$ depends on $\rho$, $p$, $s$ and $T$ and not on $R$.
The dependence on $\rho$ takes place through the local norms of $u_0$ and
$f$. The main point is that the different
constants $C_i$ appearing in this calculation do not depend on $R$.
Besides, in view of the form
of $\psi$, the first term in (\ref{3est}) (replacing $T$ by $t$, which is
possible if $0 \le t \le T$) gives, for every $0~~~~ s+1.
\end{equation}
Note that this inequality is satisfied for all $q~~N$. In this case we may select
$q$ in the range $N < q