Asymptotic Behavior of Solutions for the Cauchy Problem of a Dissipative Boussinesq-Type Equation

We consider the Cauchy problem for an evolution 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

On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity || and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case

Regularity properties and blow-up of the solutions for improved Boussinesq equations

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

Existence and decay rates for a semilinear dissipative fractional second order evolution equation

In this work we study the existence and uniqueness of solutions and decay rates to the total energy and the L2-norm of solution for a semilinear second order evolution equation with fractional damping term and under effects of a generalized rotational inertia term in the case of plate equation. This system also includes equations of Boussinesq type that model hydrodynamic problems. We show decay rates depend- ing on the fractional powers of the operators and using an asymptotic expansion of the solution to the linear problem, we prove for some cases depending on the exponents of the operators, the optimality of the decay rates