A1 Journal article (refereed)
Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton–Jacobi equations (2020)


Attouchi, A., & Souplet, P. (2020). Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton–Jacobi equations. Calculus of Variations and Partial Differential Equations, 59, Article 153. https://doi.org/10.1007/s00526-020-01810-9


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Publication details

All authors or editors: Attouchi, Amal; Souplet, Philippe

Journal or series: Calculus of Variations and Partial Differential Equations

ISSN: 0944-2669

eISSN: 1432-0835

Publication year: 2020

Publication date: 20/08/2020

Volume: 59

Article number: 153

Publisher: Springer

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1007/s00526-020-01810-9

Publication open access: Not open

Publication channel open access:

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1912.00626


Abstract

Consider the diffusive Hamilton–Jacobi equation

ut−Δu=|∇u|p+h(x) in Ω×(0,T)
with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question of the gradient blowup rate for classical solutions with p>2. We first consider the case of time-increasing solutions. For such solutions, the precise rate was obtained by Guo and Hu (2008) in one space dimension, but the higher dimensional case has remained an open question (except for radially symmetric solutions in a ball). Here, we partially answer this question by establishing the optimal estimate

C1(T−t)−1/(p−2)≤∥∇u(t)∥∞≤C2(T−t)−1/(p−2)
(1)
for time-increasing gradient blowup solutions in any convex, smooth bounded domain Ω with 2

<3. We also cover the case of (nonradial) solutions in a ball for p=3. Moreover we obtain the almost sharp rate in general (nonconvex) domains for 22, we show that more singular rates may occur for solutions which are not time-increasing. Namely, for a suitable class of solutions in one space-dimension, we prove the lower estimate ∥ux(t)∥∞≥C(T−t)−2/(p−2).


Keywords: partial differential equations

Free keywords: Hamilton–Jacobi equations


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Ministry reporting: Yes

Reporting Year: 2020

JUFO rating: 2


Last updated on 2022-19-08 at 19:57