2011/01/08 by Tai Nguyen Phuoc, Phuoc, Tai Nguyen, Laurent Veron +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1101.1576
openalex publication_date 2011/01/08 · arxiv created 2011/01/31 · arxiv updated 2011/02/01 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We study the initial value problem with unbounded nonnegative functions or measures for the equation \prttu-\Gdp u+f(u)=0 in \BBRN\ti(0,∞) where p>1, \Gdp u = div(\abs ∇ up-2 ∇ u) and f is a continuous, nondecreasing nonnegative function such that f(0)=0. In the case p>(2N)/(N+1), we provide a sufficient condition on f for existence and uniqueness of the solutions satisfying the initial data k\gd0 and we study their limit when k→∞ according f-1 and F-1/p are integrable or not at infinity, where F(s)=∫0s f(\gs)d\gs. We also give new results dealing with non uniqueness for the initial value problem with unbounded initial data. If p>2, we prove that, for a large class of nonlinearities f, any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case f(u)=u^\ga ln^\gb(u+1), where \ga>0 and \gb≥ 0.