2014/07/16 by Marie‐Françoise Bidaut‐Véron, Bidaut-Véron, Marie-Françoise, Nguyen Anh Dao +1 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1407.4442
openalex publication_date 2014/07/16 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28
Here we study the initial trace problem for the nonnegative solutions of the\nequation \u
t-
Delta u+|
nabla u|q=0 in\nQ\_\Ω,T=\Ω\×\( 0,T\) , T leqq\∞, where q>0,\nand \Ω=\ℝN, or \Ω is a smooth bounded domain of\n\ℝN and u=0 on \∂\Ω\×\( 0,T\) . We can\ndefine the trace at t=0 as a nonnegative Borel measure (\S\n,u\_0), where S is the closed set where it is infinite, and u\_0 is a\nRadon measure on \Ω backslash\S. We show that the trace is a\nRadon measure when q leqq1. For q\∈(1,(N+2)/(N+1) and any given Borel\nmeasure, we show the existence of a minimal solution, and a maximal one on\nconditions on u\_0. When \S =\\ω\∩\Ω and\n\ω is an open subset of \Ω, the existence extends to any q leqq2\nwhen u\_0\∈ L\_loc1(\Ω) and any q>1 when u\_0=0. In\nparticular there exists a self-similar nonradial solution with trace\n(\ℝN+,0), with a growth rate of order \vert x \vert\n^q\′ as \vert x \vert \→\∞ for fixed t.\nMoreover we show that the solutions with trace (\\ω,0) in\nQ\_\ℝN,T may present near t=0 a growth rate of order\nt-1/(q-1) in \ω and of order t-(2-q)/(q-1) on \∂\n\ω.\n