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

Martina Magliocca, Alessio Porretta

TL;DR

This work develops a robust well-posedness and decay theory for parabolic equations with superlinear first-order terms and bounded measurable coefficients. By focusing on the viscous Hamilton-Jacobi structure with $|H(t,x,Du)|\le \gamma|Du|^q$ for $1<q<2$, the authors establish existence and uniqueness of weak (and renormalized where needed) solutions in the optimal initial data spaces $L^\sigma(\Omega)$ (or $L^1$ in the borderline), without relying on gradient estimates from Calderón-Zygmund theory. They prove strong short-time regularizing effects, $L^\sigma-L^r$ smoothing, and exponential long-time decay in $L^\infty$, along with a comparison principle in the linear case and a sharp optimality result for the initial data class. The results significantly extend well-posedness and asymptotic behavior for parabolic problems with superlinear Hamiltonians to operators with merely bounded measurable coefficients, via elementary energy and truncation methods that rely on equi-integrability instead of gradient bounds. Together, these findings provide a broad, sharp framework for local and global behavior of solutions with unbounded data in diverse dimensions and nonlinear growth regimes.

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.

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

TL;DR

This work develops a robust well-posedness and decay theory for parabolic equations with superlinear first-order terms and bounded measurable coefficients. By focusing on the viscous Hamilton-Jacobi structure with for , the authors establish existence and uniqueness of weak (and renormalized where needed) solutions in the optimal initial data spaces (or in the borderline), without relying on gradient estimates from Calderón-Zygmund theory. They prove strong short-time regularizing effects, smoothing, and exponential long-time decay in , along with a comparison principle in the linear case and a sharp optimality result for the initial data class. The results significantly extend well-posedness and asymptotic behavior for parabolic problems with superlinear Hamiltonians to operators with merely bounded measurable coefficients, via elementary energy and truncation methods that rely on equi-integrability instead of gradient bounds. Together, these findings provide a broad, sharp framework for local and global behavior of solutions with unbounded data in diverse dimensions and nonlinear growth regimes.

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.

Paper Structure

This paper contains 12 sections, 26 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. The case $2-\frac{N}{N+1}< q<2$. Estimates in $L^{\sigma}(\Omega)$.
  3. Notions of solution
  4. A priori estimates and contraction principles.
  5. The borderline case $q=2-\frac{N}{N+1}$
  6. Regularizing effects and long time decay
  7. Regularizing effect $L^{\sigma}-L^{r}$
  8. Asymptotic behaviour
  9. The case $1<q<2-\frac{N}{N+1}$ and $L^1$-data
  10. The case $N=2$ and $N=1$.
  11. A comparison result for linear operators
  12. On the optimality of the Lebesgue class of initial data

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain in $\mathbb{R}^N, N>2$. Let $A(t,x)$ be a $N\times N$ matrix, with bounded measurable coefficients $a_{i,j}(t,x)$, which is assumed to be coercive (i.e. $A(t,x)\xi\cdot \xi\geq \alpha |\xi|^2$ for all $\xi\in \mathbb{R}^N$, for some $\alpha>0$). Let us set Then we have:

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 38 more