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| We present some uniqueness results for a class of weak solutions (the so
called BV solutions) of the Cauchy-Dirichlet problem associated to doubly
nonlinear diffusion systems. The results are established in the $BV_{t}(Q)$
space following the paper by D\'{\i}az and Padial (\emph{Uniqueness and
existence of solution in the} $BV_{t}(Q)$ \emph{space to a doubly
nonlinear parabolic problem}, Publicacions Matem\`{a}tiques, Vol. \textbf{40
}(1996)) jointly with the doubling time variable method inspired in some ideas
introduced by S.N. Kruzhkov (see, e.g., S.N. Kruzhkov, \emph{First order
quasilinear equations in several independent variables}. Math. USSR Sbornik,
\textbf{10}, No. 2, 1970, pp. 217-243). We point out that, in contrast to the
case of scalar equations, there are counterexamples showing that the
comparison principle may fail for systems of equations. In particular, we
shall consider, for instance, the problem that appear in chemical engineering
and metallurgy involving the interaction of diffusing substances with immobile
solid phases. |
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