English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the diffusive Hamilton-Jacobi equation $u_t - \Delta u = |\nablau|^p$ in a bounded planar domain with zero Dirichlet boundary condition. It isknown that, for $p>2$, the solutions to this problem can exhibit gradientblow-up (GBU) at the boundary. In this paper we study the possibility of theGBU set being reduced to a single point. In a previous work [Y.-X. Li, Ph.Souplet, 2009], it was shown that single point GBU solutions can be constructedin very particular domains, i.e.~locally flat domains and disks. Here, we provethe existence of single point GBU solutions in a large class of domains, forwhich the curvature of the boundary may be nonconstant near the GBU point.  Our strategy is to use a boundary-fitted curvilinear coordinate system,combined with suitable auxiliary functions and appropriate monotonicityproperties of the solution. The derivation and analysis of the parabolicequations satisfied by the auxiliary functions necessitate long and technicalcalculations involving boundary-fitted coordinates.

Single-point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equation in domains with non-constant curvature