Global existence and nonexistence of the initial–boundary value problem for the dissipative Boussinesq equation

Blow-up in damped abstract nonlinear equations

“…Most of them are results for blow up of solutions, see [29,25,16]. Similar analysis, for E 0 ≥ d, have been performed to prove the same qualitative properties for other equations, see [14,28] to cite just some of the most influential papers on the subject, and see [4,19,20,26,30,31,35,36,37] and references therein for some recent works. In [6] a numerical study for the Cauchy problem of the focusing cubic nonlinear Klein-Gordon equation in three dimensions for radial initial data shows that for high energies, the qualitative behavior seems to be much more complicated than for E < d, and more research is required to find a threshold between globality and blow up of solutions.…”

“…Moreover, by Corollary 1, for every positive initial energy E 0 , there exists initial data such that imply the nonexistence of global solutions in the norm of H. There are several results in the literature showing blow up for large positive values of the initial energy for equations of the type (GB) * . They consider either δ = 0, see [16,4,36], or a linear damping term, which can be either weak or strong, see [25,26,29,30,31]. In most of them, the blow up is proved under the assumption (2) and if the initial energy satisfies an inequality of the type (17) E…”

Optimal decay rates and small global solutions to the dissipative Boussinesq equation

“…From the above argument, we find when taking = 1 in (11), = 2, = 1, = 1 in Equation (12), = 0, = 1 in Equation (13), respectively, they become Equation (8) with = 0. However, by comparing the results in the previous studies [17][18][19] with this paper, we find that the decay rate of the solutions, showed in this paper, are different from the ones in the previous studies.…”

“…Obviously, Equation (1) is the general multidimensional form of Equation (7). Wang and Su studied the existence and uniqueness of global solutions, the finite time blow-up of solutions, and the long-time behavior of solutions for the initial-boundary value problem of Equation (1) in bounded domains, 13,14 and the authors also considered the existence and nonexistence of the global solutions for the Cauchy problem of Equation (1) without the decay estimates of the solutions. 15 The main purpose of the paper is to study the asymptotic behavior and the optimal decay rate of the linearized equation of Equation (1) and the global existence and the optimal decay estimates of small solutions for the nonlinear problem (1), (2).…”

Blow-up for the sixth-order multidimensional generalized Boussinesq equation with arbitrarily high initial energy

“…Castro et al [21] considered 2D Boussinesq equations with a velocity damping term in a strip with impermeable walls and obtained the asymptotic stability for a specific type of perturbations of a stratified solution by using a suitably weighted energy space combined with linear decay, Duhamel's formula, and "bootstrap" arguments. Wang and Su [22] studied multidimensional dissipative Boussinesq equations and obtained the sufficient conditions for global solutions and finite time blow-up solutions respectively with three different cases of initial energy. Particularly, by using some new methods and some analysis techniques, the authors gave the novel contribution for the blow-up result with initial energy at supercritical initial energy.…”