Existence and uniqueness of solutions to parabolic equations with superlinear Hamiltonians

TL;DR

This paper proves the existence and uniqueness of viscosity solutions for a broad class of parabolic quasilinear equations with superlinear, nonconvex Hamiltonians, using classical techniques and Lipschitz estimates.

Contribution

It extends the theory of viscosity solutions to include nonconvex Hamiltonians with superlinear growth, providing a general existence and uniqueness result.

Findings

Established existence and uniqueness of solutions.

Applied classical techniques and Lipschitz estimates.

Handled nonconvex Hamiltonians with superlinear growth.

Abstract

We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear 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 proved in [S.N. Armstrong and H.V. Tran, Viscosity solutions of general viscous Hamilton-Jacobi equations, Math. Ann., 361 (2015)], as well as on viscosity solution arguments.