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A doubly critical degenerate parabolic problem
Author(s) -
Winkler Michael
Publication year - 2004
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.487
Subject(s) - mathematics , bounded function , degenerate energy levels , eigenfunction , eigenvalues and eigenvectors , dirichlet problem , geodetic datum , laplace operator , mathematical analysis , dirichlet distribution , pure mathematics , boundary value problem , physics , cartography , quantum mechanics , geography
It is shown that the Dirichlet problem for$$u_t=u^p(\Delta u + u)\quad {\rm in}\enspace \Omega \times (0,T),\quad p>0$$ where Ω⊂ℝ n is critical in that it has first eigenvalue one, is globally solvable for any continuous positive initial datum vanishing at ∂ Ω. Moreover, for p <3 all solutions are bounded and tend to some nonnegative eigenfunction of the Laplacian as t →∞, while if p ⩾3 then there are both bounded and unbounded solutions. Finally, it is shown that unlike the case p ∈[0,1), all steady states are unstable if p ⩾1. Copyright © 2004 John Wiley & Sons, Ltd.