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

Abstract

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}

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^{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 $L^{1}\left( \Omega \right) .$