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This work is devoted to the small amplitude solutions for the initial value problem of the multidimensional dissipative Boussinesq equation. We firstly derive the pointwise estimates of the fundamental solutions by the energy method in the Fourier space. We give the asymptotic profiles of solutions to the corresponding linear problem to get the optimal decay rate for the Ḣs‐norm of solutions in all space dimensions. Under smallness assumptions on the initial data, we study the global existence and uniqueness of solutions by the contractive mapping principle in the solution spaces with time weighted norm. [ABSTRACT FROM AUTHOR]

Abstract This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a Boussinesq-type equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle.

We consider the Cauchy problem for a Boussinesq-type equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle. [ABSTRACT FROM PUBLISHER]

We investigate the initial–boundary value problem of n -dimensional ( n ≥ 1 ) dissipative Boussinesq equation. The global existence, uniqueness and stability of the strong solutions are obtained by means of the Faedo–Galërkin method. We discuss the long-time behavior of the strong solutions so as to prove the existence of global compact attractor and exponential attractor. [ABSTRACT FROM AUTHOR]

A class of multidimensional dissipative Boussinesq equations is considered. Under three different cases of initial energy, we prove the existence and uniqueness of global solutions in the energy space and provide sufficient conditions of finite time blow up solutions. The sharp result is given as E ( 0 ) < d . Particular attention is paid to initial energy at supercritical initial energy ( E ( 0 ) > d ) level which is novel contribution of this paper, and new methods will be introduced and some analysis techniques are needed. In addition, the asymptotic behavior of the global solutions is studied. [ABSTRACT FROM AUTHOR]