Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process

N. I. Kavallaris, A. A. Lacey, D. E. Tzanetis

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation u t=?u+?f(u)/(?Of(u)dx) 2,O?RN,N=1,2. Under specific conditions on f, there exists a ?* such that for each 0<?<?* there corresponds a unique steady-state solution and u=u(x,t;?) is a global in time-bounded solution, which tends to the unique steady-state solution as t?8 uniformly in x. Whereas for ?=?* there is no steady state and if ?>?* then u blows up globally. Here, we show that when (a) N=1,O=(-1,1) and f(s)>0,f'(s)<0,s=0, or (b) N=2,O=B(0,1) and f(s)=e -s, the solution u*=u(x,t;? *) is global in time and diverges in the sense ||u *(·,t)||8?8, as t?8. Moreover, it is proved that this divergence is global i.e. u*(x,t)?8 as t?8 for all x?O. The asymptotic form of divergence is also discussed for some special cases. © 2004 Elsevier Ltd. All rights reserved.

Original languageEnglish
Pages (from-to)787-812
Number of pages26
JournalNonlinear Analysis: Theory, Methods and Applications
Volume58
Issue number7-8
DOIs
Publication statusPublished - Sep 2004

Keywords

  • Asymptotic behaviour
  • Comparison methods
  • Global-unbounded solutions
  • Non-local parabolic problems

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