In this paper, we study the Cauchy problem for linear and nonlinear
Boussinesq type equations that include the general differential operators.
First, by virtue of the Fourier multipliers, embedding theorems in Sobolev
and Besov spaces, the existence, uniqueness, and regularity properties of
the solution of the Cauchy problem for the corresponding linear equation
are established. Here,
L
p
-estimates for a~solution with respect to
space variables are obtained uniformly in time depending on the given data
functions. Then, the estimates for the solution of linearized equation and
perturbation of operators can be used to obtain the existence, uniqueness,
regularity properties, and blow-up of solution at the finite time of the
Cauchy for nonlinear for same classes of Boussinesq equations. Here, the
existence, uniqueness,
L
p
-regularity, and blow-up properties of the
solution of the Cauchy problem for Boussinesq equations with differential
operators coefficients are handled associated with the growth nature of
symbols of these differential operators and their interrelationships. We
can obtain the existence, uniqueness, and qualitative properties of
different classes of improved Boussinesq equations by choosing the given
differential operators, which occur in a wide variety of physical systems.
On the Cauchy problem for Schrödinger–improved Boussinesq equations