Abstract. We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation u t = div a(u, Du), where a(z, ξ ) = ∇ ξ f (z, ξ ), and f is a convex function of ξ with linear growth as ξ → ∞, satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.
Abstract. In this paper we study existence and uniqueness for solutions of the nonlocal diffusion equation with Neumann boundary conditionsand for solutions of its local counterpartWe consider 1 ≤ p < ∞ and g ≥ 0. We pay special attention to the case in which g vanishes somewhere in Ω, even in a set of positive measure.
We introduce a new concept of solution for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. Using Kruzhkov's method of doubling variables both in space and time we prove uniqueness and a comparison principle in L 1 for these solutions. To prove the existence we use the nonlinear semigroup theory. (2000): 35K55, 35K65, 47H06, 47H20
Mathematics Subject Classification
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