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

Kavallaris, NI; Lacey, AA; Tzanetis, DE

dc.contributor.author	Kavallaris, NI	en
dc.contributor.author	Lacey, AA	en
dc.contributor.author	Tzanetis, DE	en
dc.date.accessioned	2014-03-01T01:20:34Z
dc.date.available	2014-03-01T01:20:34Z
dc.date.issued	2004	en
dc.identifier.issn	0362-546X	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/15966
dc.subject	Asymptotic behaviour	en
dc.subject	Comparison methods	en
dc.subject	Global-unbounded solutions	en
dc.subject	Non-local parabolic problems	en
dc.subject.classification	Mathematics, Applied	en
dc.subject.classification	Mathematics	en
dc.subject.other	Diffusion	en
dc.subject.other	Electric conductivity	en
dc.subject.other	Electric currents	en
dc.subject.other	Electric resistance	en
dc.subject.other	Heating	en
dc.subject.other	Set theory	en
dc.subject.other	Asymptotic behavior	en
dc.subject.other	Comparison methods	en
dc.subject.other	Global-unbounded solutions	en
dc.subject.other	Non-local parabolic problems	en
dc.subject.other	Boundary value problems	en
dc.title	Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process	en
heal.type	journalArticle	en
heal.identifier.primary	10.1016/j.na.2004.04.012	en
heal.identifier.secondary	http://dx.doi.org/10.1016/j.na.2004.04.012	en
heal.language	English	en
heal.publicationDate	2004	en
heal.abstract	We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation u(t) = Deltau + lambdaf(u)/(integral(Omega)f(u)dx)(2), Omega subset of R-N, N = 1, 2. Under specific conditions on f, there exists a lambda* such that for each 0 < lambda < lambda* there corresponds a unique steady-state solution and u = u(x, t; lambda) is a global in time-bounded solution, which tends to the unique steady-state solution as t --> infinity uniformly in x. Whereas for lambda greater than or equal to lambda* there is no steady state and if lambda > lambda* then u blows up globally. Here, we show that when (a) N = 1, Omega = (-1, 1) and f(s) > 0, f'(s) < 0, s greater than or equal to 0, or (b) N = 2, Omega = B(0, 1) and f(s) = e(-s), the solution u* = u(x, t; lambda) is global in time and diverges in the sense \\u((.),t)\\(infinity) --> infinity, as t --> infinity. Moreover, it is proved that this divergence is global i.e. u*(x, t) --> infinity as t --> infinity for all x is an element of Q. The asymptotic form of divergence is also discussed for some special cases. (C) 2004 Elsevier Ltd. All rights reserved.	en
heal.publisher	PERGAMON-ELSEVIER SCIENCE LTD	en
heal.journalName	Nonlinear Analysis, Theory, Methods and Applications	en
dc.identifier.doi	10.1016/j.na.2004.04.012	en
dc.identifier.isi	ISI:000223830300004	en
dc.identifier.volume	58	en
dc.identifier.issue	7-8	en
dc.identifier.spage	787	en
dc.identifier.epage	812	en