The paper is devoted to the proof of existence and uniqueness of entropy solutions of the Cauchy problem for the quasilinear parabolic equation \[ u_t= \text{div}\, {\mathbf a}(u, Du) \] with initial condition \(u_0 \in BV(\mathbb R^N)\), \(u_0 \geq 0\), and \({\mathbf a}(z, \xi)=\nabla_\xi f(z, \xi)\), where \(f\) is convex in the second variable, with linear growth when \(\|\xi\| \to +\infty\) and satisfies other additional assumptions which, in turn, allow to apply the achieved results to many physical models (cf. Remarks 1 and 2). The existence is proved by means of Crandall-Liggett’s scheme and uniqueness by means of Kruzhkov’s technique of doubling variables. In order to use the Kruzhokov’s method the author develops a functional calculus, relying also on some recent lower semicontinuity results for integral functionals defined on \(BV\).