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Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents by Victor A. Galaktionov; John R. King is a Mathematics article available to read on EtoBox.

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We describe special asymptotic structures of solutions of the semilinear heat equation in the unit ball O 1⁄4 fr 1⁄4 jxjo1gCR N in dimensions NX3 with positive symmetric initial data u 0 ðrÞ: It is known (J. Differential Equations 54 (1984) 97) that in the critical Sobolev case p 1⁄4 p s 1⁄4 ðN þ 2Þ=ðN À 2Þ the Cauchy problem with specially chosen initial data u 0 admits global unbounded solutions (GUSs) uðÁ; tÞ; which are uniformly bounded in L 1 ðOÞ but are not bounded in L N ðOÞ for all t40: In the radial geometry, we establish the following asymptotic behaviour of such solutions as t-N: and jjuðÁ; tÞjj N Bg 0 t ðNÀ2Þ=2ðNÀ4Þ for NX5; where g 0 1⁄4 g 0 ðNÞ40 is a constant independent of initial data. The large-time behaviour is not self-similar and is obtained by matching of inner and outer asymptotic expansions. The phenomenon of GUSs is shown to be a common feature of a number of quasilinear and fully nonlinear parabolic equations with scaling invariant operators in the critical Sobolev case.

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Author

Victor A. Galaktionov; John R. King

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-0396)

Published

2003

Language

EN

Field

Mathematics (Physical Sciences)