Critical convective-type equations on a half-line

Elena I. Kaikina

International Journal of Mathematics and Mathematical Sciences, 2006, vol. 2006, 1-24

Abstract:

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equations u t + â„• ( u , u x ) + ( a n ∂ x n + a m ∂ x m ) u = 0 , ( x , t ) ∈ â„ + × â„ + , u ( x , 0 ) = u 0 ( x ) , x ∈ â„ + , ∂ x j − 1 u ( 0 , t ) = 0 for j = 1 , … , m / 2 , where the constants a n , a m ∈ â„ , n , m are integers, the nonlinear term â„• ( u , u x ) depends on the unknown function u and its derivative u x and satisfies the estimate | â„• ( u , v ) | ≤ C | u | Ï | v | σ with σ ≥ 0 , Ï â‰¥ 1 , such that ( ( n + 2 ) / 2 n ) ( σ + Ï âˆ’ 1 ) = 1 , Ï â‰¥ 1 , σ ∈ [ 0 , m ) . Also we suppose that ∫ â„ + x n / 2 â„• d x = 0 . The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem above-mentioned. We find the main term of the asymptotic representation of solutions in critical case. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in critical convective case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:084972

DOI: 10.1155/IJMMS/2006/84972

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