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
JYU authors or editors
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
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
Contributing organizations
Related projects
- Regularity issues for the normalized p-Laplacian and more general parabolic equations in non-divergence form.
- Attouchi, Amal
- Academy of Finland
Ministry reporting: Yes
Reporting Year: 2020
JUFO rating: 2
