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

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 norm of the solution n, ¨.with sufficiently small and regular initial data decays to zero like ty1r3. 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.