Existence and Uniqueness of a Solution to the Cauchy Problem for the Damped Boussinesq Equation

The Cauchy problem for the nonlinear viscous Boussinesq equation in the $L^q$ framework

“…Actually, the corresponding abstract problem (but with different nonlinear terms) for (2) has been studied by [2,3], where global (in time) well-posedness and time-decay estimates were derived. Facing exactly with the Cauchy problem for (2), the authors in [43,42] proved local (in time) wellposedness by employing successive approximations. Later, some global (in time) existence and blowup results for (2) in 1D were established in [45].…”

“…Our next aim is to turn the proof of the Lipschitz condition (38). Our proof is based on the similar approach to the proof of the obtained inequality (42). Indeed, we may estimate Analogously to the proof of ( 39) and ( 40…”

Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation

“…Aassila and Guesmia [11] further obtained the energy decay of the above-mentioned solutions. Obviously, both conditions (9) and (10) imply that the growth order of the nonlinear term g(s) is not more than 1.…”

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

“…H s−2 , and |x|(−Δ) −1 u 1 ∈ L 1 , and v be the solution of the linear problem (8), (2). Then, there exist constants C 1 > 0, C 2 > 0 and a sufficiently large time t 1 such that for all t ≥ t 1 ,…”