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Global existence for degenerate parabolic equations with a non‐local forcing
Author(s) -
Anderson Jeffrey R.,
Deng Keng
Publication year - 1997
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19970910)20:13<1069::aid-mma867>3.0.co;2-y
Subject(s) - mathematics , monotone polygon , degenerate energy levels , reaction–diffusion system , forcing (mathematics) , term (time) , convection , convection–diffusion equation , diffusion , parabolic partial differential equation , mathematical analysis , partial differential equation , geometry , physics , mechanics , quantum mechanics , thermodynamics
We establish local existence and comparison for a model problem which incorporates the effects of non‐linear diffusion, convection and reaction. The reaction term to be considered contains a non‐local dependence, and we show that local solutions can be obtained via monotone limits of solutions to appropriately regularized problems. Utilizing this construction, it is further shown that, under conditions of either ‘weak reaction’ or ‘sufficiently small’ initial mass, solutions exist for all time. Finally, we provide an alternative analysis of global existence and investigate blow up in finite time for the case of power law diffusion and convection. These results show the extent to which the assumption of weak reaction may be relaxed and still obtain global existence. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.