An existence and uniqueness result for flux limited diffusion equations (Q653887)

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 \(BV\).