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This paper deals with the Cauchy problem for the quasilinear parabolic equations with arbitrary degeneracy. We give a new and natural definition of BV entropy solution. The uniqueness of the natural BV entropy solutions is obtained. In order to prove the uniqueness, the discontinuity conditions for these solutions are established. Also, we construct an example of the natural BV entropy solution.

Uniqeness of <i>BV</i> Entropy Solutions for High Dimensional Quasilinear Parabolic Equations with Arbitrary Degeneracy