On the uniqueness of the solution of the evolution dam problem

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The purpose here is to investigate some aspects of the evolution dam problem $\partial(\chi+\alpha u)/\partial t=\Delta u+{\rm div}(\chi e)$ in $Q$, $\chi\in H(u)$ in $Q$, $u\geq 0$ in $Q$, $u=\phi$ in $\Sigma_2$, $\partial u/\partial v+\chi ev\leq 0$ on $\Sigma_2\cap \{\phi=0\}$, $e\in {\bf R}^n$, $e=(0,\cdots, 0,1)$, $\partial u/\partial v+\chi ev=0$ on $\Sigma_1$. Here $H$ is the Heaviside maximal graph $H(s)=1$ if $s>0$, $H(s)=[0,1]$ if $s=0$, $\alpha$ is the storativity constant: $\alpha=0$ if the fluid is incompressible and $\alpha>0$ if the fluid is compressible. Also $Q=\Omega\times (0,T)$ where $\Omega$ is a bounded domain in ${\bf R}^n$ ($\Omega$ represents a porous medium separating a finite number of reservoirs) with Lipschitz boundary $G$ where $G$ is divided into two parts: $G_1$, which is the impervious part, and $G_2$, which is the pervious part, with the following assumptions: $G_2$ is an open subset of $G$, $G_2\not=\emptyset$, $G_2\cap G_1=\emptyset$, $G_2\cup G_1=G$ and $e\nu\leq 0$ a.e. on $G_1$; $\Sigma_i$ denotes $G_i\times (0,T)$, $i=1,2$. The pressure at $\Sigma_2$ is known and is represented by a nonnegative function $\phi\in C^{0,1}(\overline Q)$. The above problem can be completed by prescribing an initial value to $\chi+\alpha u$. The existence of a weak solution of the evolution dam problem is well known; however, many properties of the solutions of the initial value dam problem, in particular, comparison and uniqueness theorems, continue to be open for a general domain $\Omega$. The results of this paper give precise information about the comparison (Theorem 4.7) and the uniqueness (Corollary 4.9) statements. The key point to prove comparison and uniqueness results is that the function $h\mapsto \chi(x',x_n+h, t-h)$, $x'\in{\bf R}^{n-1}$, is a.e. nonincreasing