Critical global asymptotics in higher-order semilinear parabolic equations

Victor A. Galaktionov

International Journal of Mathematics and Mathematical Sciences, 2003, vol. 2003, 1-17

Abstract:

We consider a higher-order semilinear parabolic equation u t = − ( − Δ ) m u − g ( x , u ) in ℝ N × ℝ + , m > 1 . The nonlinear term is homogeneous: g ( x , s u ) ≡ | s | p − 1 s g ( x , u ) and g ( s x , u ) ≡ | s | Q g ( x , u ) for any s ∈ ℝ , with exponents P > 1 , and Q > − 2 m . We also assume that g satisfies necessary coercivity and monotonicity conditions for global existence of solutions with sufficiently small initial data. The equation is invariant under a group of scaling transformations. We show that there exists a critical exponent P = 1 + ( 2 m + Q ) / N such that the asymptotic behavior as t → ∞ of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solution b ( x , t ) = t − N / 2 m f ( x t − 1 / 2 m ) of the parabolic operator ∂ / ∂ t + ( − Δ ) m , so that for t ≫ 1 , u ( x , t ) = C 0 ( ln t ) − N / ( 2 m + Q ) [ b ( x , t ) + o ( 1 ) ] , where C 0 is a constant depending on m , N , and Q only.

Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:406190

DOI: 10.1155/S0161171203210176

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