On the generalized Burgers equation

The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u t + g ( t , u ) u x + f ( t , u ) = εu xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max | u x | independent of ε and t and give an exa...

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Bibliographic Details
Published inNonlinear differential equations and applications Vol. 17; no. 4; pp. 437 - 452
Main Author Tersenov, Alkis S.
Format Journal Article
LanguageEnglish
Published Basel SP Birkhäuser Verlag Basel 01.08.2010
Subjects
Analysis
Mathematics
Mathematics and Statistics
35F25
Boundary value problems
35K60
35F30
A priori estimates
Global solvability
Semilinear parabolic equations
35K55
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Summary:The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u t + g ( t , u ) u x + f ( t , u ) = εu xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max | u x | independent of ε and t and give an example demonstrating the optimality of this condition. Based on this estimate we prove the global existence of a unique classical solution of the problem and investigate the behavior of this solution for ε → 0 and t → + ∞. The Cauchy problem for this equation is considered as well.
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-010-0061-6