Ricerc@Sapienza

We give a proof of existence and uniqueness of viscosity solutions to parabolic quasi- linear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on classical techniques for uniformly parabolic quasilinear equations and on the Lipschitz estimates provided in [S. N. Armstrong and H. V. Tran, Viscosity solutions of general viscous Hamilton–Jacobi equations, Math. Ann. 361 (2015) 647–687], as well as on viscosity solution arguments.

We consider the partial differential equation

u−f=div(u^m ∇u/|∇u|)
with f nonnegative and bounded and m∈R. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ℋ^{N−1}-Hausdorff measure. Results and proofs extend to more general nonlinearities.

We consider the Cauchy problem for two prototypes of flux-saturated diffusion equations. In arbitrary space dimension, we give an optimal condition on the growth of the initial datum which discriminates between occurrence or nonoccurrence of a waiting time phenomenon. We also prove optimal upper bounds on the waiting time. Our argument is based on the introduction of suitable families of subsolutions and on a comparison result for a general class of flux-saturated diffusion equations.