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

Abstract

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity γ|u| p 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.

On the existence of global solutions for a nonlinear Klein-Gordon equation

“…We can also state the following properties of D σ , which can be proved easily. The following theorems show the invariance of I σ under the flow of (2), (3) for 0 < E (0) < d and E (0) = d, respectively, and can be proved by contradiction as in [20]. Now, we give a lemma for σ > 1, which states similar results to Lemmas 2.1, 2.2, and can be proved similarly.…”

“…A new functional which includes both the initial displacement u 0 and initial velocity u 1 will be constructed for the case of high energy initial data. Functionals depending on u 0 and u 1 are introduced for the first time in [8] and then they were successfully applied for proving the global existence to some Boussinesq-type equations in [20][21][22].…”