On $L^{\infty }$ and $L^{2}$ Bounds for Weak Solutions of Time Flowsfor Certain Functionals of Linear Growth

We use recent approximation results in $BV$ space to derive $L^{\infty }$ bounds for $u,$ $L^{\infty }$ bounds in time for the $BV$ seminorm $\int |Du| $ of $u,$ and $L^{2}$ bounds for $u_{t}$ for the weak solution $u\in C([0,\infty );L^{2}\left( \Omega \right) \cap BV\left( \Omega \right) ),$ $% \Omega \subset %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{N}$ open and bounded, of the time flow \begin{equation*} \frac{\partial u}{\partial t}=\func{div}\nabla _{p}\varphi (x,Du)-\lambda (u-u_{0}),\text{ }\lambda >0,\text{ }u(0,x)=u_{0}. \end{equation*}% We assume Neumann boundary condition and $\varphi (x,p)$ is in a class of linear growth functions in $p.$ Importantly, $\varphi (\cdot ,p)\in L^{1}\left( \Omega \right) $ in contrast to the classical results stated in \cite{ACM} where $\varphi $ has a continuity assumption in the $x$ variable. We also use the convergence of the solution above to derive an $L^{\infty }$ bound for the solution $u^{\ast }$ to the corresponding stationary problem, since $u(t)\rightarrow u^{\ast }$ in $L^{1}\left( \Omega \right) .$