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.
|Number of pages||26|
|Journal||Nonlinear Analysis: Theory, Methods and Applications|
|Publication status||Published - Sep 2004|
- Asymptotic behaviour
- Comparison methods
- Global-unbounded solutions
- Non-local parabolic problems