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Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution
Author(s) -
Bernard Guy
Publication year - 2009
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/mma.1235
Subject(s) - mathematics , reaction–diffusion system , upper and lower bounds , interval (graph theory) , space (punctuation) , heat equation , diffusion , initial value problem , mathematical analysis , sequence (biology) , diffusion equation , combinatorics , thermodynamics , chemistry , physics , philosophy , service (business) , linguistics , biochemistry , economy , economics
A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation − u t +Δ u + u p =0 in ℝ n ×[0, ∞), for exponents p ⩾3 and space dimensions n ⩾3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright© 2009 John Wiley & Sons, Ltd.