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Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process
University of Athens, GRC.ORCID iD: 0000-0002-9743-8636
Heriot-Watt University, GBR.
University of Athens, GRC.
2004 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 58, no 7-8, p. 787-812Article in journal (Refereed) Published
Abstract [en]

We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation ut=Δu+λf(u)/(∫Ωf(u)dx)2,Ω⊂RN,N=1,2. Under specific conditions on f, there exists a λ∗ such that for each 0<λ<λ∗ there corresponds a unique steady-state solution and u=u(x,t;λ) is a global in time-bounded solution, which tends to the unique steady-state solution as t→∞ uniformly in x. Whereas for λ⩾λ∗ there is no steady state and if λ>λ∗ then u blows up globally. Here, we show that when (a) N=1,Ω=(−1,1) and f(s)>0,f′(s)<0,s⩾0, or (b) N=2,Ω=B(0,1) and f(s)=e−s, the solution u∗=u(x,t;λ∗) is global in time and diverges in the sense ||u∗(·,t)||∞→∞, as t→∞. Moreover, it is proved that this divergence is global i.e. u∗(x,t)→∞ as t→∞ for all x∈Ω. The asymptotic form of divergence is also discussed for some special cases.

Place, publisher, year, edition, pages
Elsevier, 2004. Vol. 58, no 7-8, p. 787-812
Keywords [en]
Non-local parabolic problems, Global-unbounded solutions, Comparison methods, Asymptotic behaviour
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-88627DOI: 10.1016/j.na.2004.04.012ISI: 000223830300004Scopus ID: 2-s2.0-4243097094OAI: oai:DiVA.org:kau-88627DiVA, id: diva2:1638758
Available from: 2022-02-17 Created: 2022-02-17 Last updated: 2022-11-21Bibliographically approved

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Kavallaris, Nikos I.

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