| Abstract: |
| We will discuss the regularizing effect induced by superlinear terms in some class of Elliptic and Parabolic Equations.
Our model equations are
\begin{equation*}\label{P}
u_t-\Delta u=g(u)|\nabla u|^q+f \qquad\text{in }(0,T)\times \Omega,
\end{equation*}
and
\begin{equation*}\label{E}
-\Delta u=g(u)|\nabla u|^q+f\qquad \text{in }\Omega,
\end{equation*}
being $\Omega\subset \mathbb{R}^N$ for $N\ge 2$, $g(u)>0$ and $q < 2$.
We assume that the term
\begin{equation*}%\label{super}
g(u)|\nabla u|^q
\end{equation*}
behaves in a superlinear way.
Roughly speaking, if $g(u)\equiv \text{const}.$, then we are asking for $1 < q < 2$. In the more general case $g(u)\not\equiv \text{const}.$, the $q$ threshold is influenced by this perturbation term and the superlinear $q$ range depends on its growth. An important remark on this kind of problems concerns the data assumptions, which have to satisfy well precises compatibility conditions in order to have existence of solutions. \
We will show that, under certain growth assumptions on $g(u)$, we can relax the regularity needed on the data w.r.t. the case $g(u)=\text{const}.$. |
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