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The long-time behaviour of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions

Published online by Cambridge University Press:  14 November 2011

Eva Fašangová  and
Eduard Feireisl
Eva Fašangová
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 00 Prague, Czech Republic
Eduard Feireisl
Affiliation:
Institute of Mathematics AV ČR, Žitná 25, 115 67 Prague. Czech Republic
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Abstract

For a non-negative function ū(x), we study the long-time behaviour of solutions of the heat equation

with the Dirichlet or Neumann boundary conditions at x = 0. We find a critical parameter λD > 0 such that the solution subjected to the Dirichlet boundary condition tends to a spatially localized wave travelling to infinity in the space variable. On the other hand, there exists a λN > 0 such that the corresponding solution of the Neumann problem converges to a non-trivial strictly positive stationary solution. Consequently, the dynamics is considerably influenced by the choice of boundary conditions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 2 , 1999 , pp. 319 - 329
Copyright
Copyright © Royal Society of Edinburgh 1999

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