TY - JOUR
T1 - Global existence and divergence of critical solutions of a non-local parabolic problem in Ohmic heating process
AU - Kavallaris, N. I.
AU - Lacey, A. A.
AU - Tzanetis, D. E.
PY - 2004/9
Y1 - 2004/9
N2 - 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.
AB - 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.
KW - Asymptotic behaviour
KW - Comparison methods
KW - Global-unbounded solutions
KW - Non-local parabolic problems
UR - http://www.scopus.com/inward/record.url?scp=4243097094&partnerID=8YFLogxK
U2 - 10.1016/j.na.2004.04.012
DO - 10.1016/j.na.2004.04.012
M3 - Article
VL - 58
SP - 787
EP - 812
JO - Nonlinear Analysis: Theory, Methods and Applications
JF - Nonlinear Analysis: Theory, Methods and Applications
SN - 0362-546X
IS - 7-8
ER -