Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

We consider the degenerate parabolic equation delta(t)u + divfx(u) = div(div(A(x)(u))), x is an element of M, t >= 0 on a smooth, compact, d-dimensional Riemannian manifold (M, g). Here, for each u is an element of R, x -> f(x)(u) is a vector field and x -> Ax(u) is a (1, 1)-tensor field on M such that u -> (Ax(u)g g), 4 is an element of TIM, is non decreasing with respect to u. The fact that the notion of divergence appearing in the equation depends on the metric g requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem. (C) 2017 Elsevier Inc. All rights reserved.