2009/07/04 by Marina Ghisi, Massimo Gobbino, Ghisi, Marina +1 · 1 citation
Engineering · Mathematics · Medicine · #35B05 #35K55 #35K65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Guidance and Control Systems #Mathematical and Theoretical Epidemiology and Ecology Models #Numerical methods for differential equations #math.AP #msc:35B05 #msc:35K55 #msc:35K65
paper · pdf · doi:10.48550/arxiv.0907.0772
38 pages, 3 figures
arxiv created 2009/07/04 · openalex publication_date 2009/07/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Cauchy problem for the Perona-Malik equation in an open subset of Rn, with Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C1 solution if the initial condition is transcritical, namely when the norm of the gradient of the initial condition is smaller than 1 in some region, and larger than 1 in some other region In this paper we show that this result cannot be extended to higher dimension. We show indeed that for n >= 2 the problem admits radial solutions of class C2,1 with a transcritical initial condition.