On L∞ and L2 Bounds for Weak Solutions of Time Flowsfor Certain Functionals of Linear Growth

Abstract

We use recent approximation results in $BV$ space to derive $L^{\mathrm{\infty }}$

bounds for $u,$ $L^{\mathrm{\infty }}$ bounds in time for the $BV$ seminorm $\int

|Du| $of$u,$and$L^{2}$boundsfor$u_{t}$fortheweaksolution$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 $\phi (x,p)$ 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 $\phi$ has a continuity assumption in the $x$ variable.

We also use the convergence of the solution above to derive an $L^{\mathrm{\infty }}$

bound for the solution $u^{\ast }$ to the corresponding stationary problem,

since $u(t)\to u^{\ast }$ in $L^1\left(\mathrm{\Omega }\right).$