Reduced measures associated to parabolic problems

Abstract

We study the existence and the properties of the reduced measures for the parabolic equations ${\partial }_{t}u-\Delta u+g(u)=0$ in $\Omega \times (0,\infty )$ subject to the conditions ($P$): $u=0$ on $\partial \Omega \times (0,\infty )$, $u(x,0)=\mu$ and ($P^{\prime }$): $u={\mu }^{\prime }$ on $\partial \Omega \times (0,\infty )$, $u(x,0)=0$ where $\mu$ and ${\mu }^{\prime }$ are positive Radon measures and $g$ a continuous nondecreasing function