Local and global time decay for parabolic equations with super linear first order terms

TL;DR

This paper investigates parabolic equations with superlinear first order terms, establishing well-posedness, regularization, and decay estimates for solutions with unbounded initial data, using elementary methods and minimal regularity assumptions.

Contribution

It introduces a novel approach that handles nonlinear operators with merely measurable and bounded coefficients, avoiding higher-order gradient estimates.

Findings

Well-posedness in optimal Lebesgue spaces for unbounded initial data

Regularizing effects and decay estimates for solutions

Applicability to nonlinear operators with minimal regularity assumptions

Abstract

We study a class of parabolic equations having first order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.