Time Decay of Solutions for Generalized Boussinesq Equations in Two Space Dimensions

✍ Scribed by Akmel Dé Godefroy

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 201 KB
• Volume: 237
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1999.6470

✦ Synopsis

In this article, we study the Cauchy problem of generalized Boussinesq equations. We prove the local existence in time in Sobolev and weighted Sobolev space through Fourier transforms. Then our main result is to prove that the supremum Ž . norm of the solution n, ¨with sufficiently small and regular initial data decays to zero like t y1 r3 . The proof of this result is based on the analysis of the linear part of these Boussinesq equations. After diagonalization of the symbol of the matrix operator associated with the linearized equations, it appears that the components of the eigenvectors associated with the eigenvalues of this matrix valued symbol play a significant role in the difficulties we encountered in our study. ᮊ 1999 Academic Press

. with initial data n x, 0 , ¨x, 0 s n x , ¨x , and where p ) 0 is an 0 0 integer. n represents the longitudinal velocity and ¨represents the trans-2 2

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verse velocity, with n, ¨decaying to 0 as x q x ª ϱ. ␤ ) 0 measures 1 2 the dispersive effect and ⑀ ) 0 and ␥ ) 0 measure, respectively, the * This paper is in great part, a portion of the author's Ph.D. thesis, written at the