The power series method for nonlocal and nonlinear evolution equations

Abstract

Please cite this article in press as: R.F. Barostichi et al., The power series method for nonlocal and nonlinear evolution equations, J. Math. Anal. Appl. (2016), http://dx.Abstract. The initial value problem for a 4-parameter family of nonlocal and nonlinear evolution equations with data in a space of analytic functions is solved by using a power series method in abstract Banach spaces. In addition to determining the power series expansion of the solution, this method also provides an estimate of the analytic lifespan expressed in terms of the norm of the initial data, thus establishing an abstract Cauchy-Kovalevsky type theorem for these equations.

Local and global analyticity for $\mu$-Camassa-Holm equations

“…The second inequality is easy to prove. We give a proof of the first in order to clarify that the assumption δ ≤ 1 in [1,2] is superfluous. The present author thinks that the authors of [1,2] wrote δ ≤ 1 not because they really needed it for the omitted proof but for the sole reason that they were interested only in 0 < δ ≤ 1.…”

“…We recall some basic facts about the autonomous Ovsyannikov theorem. Among many versions, we adopt the one in [1,2].…”

“…where k = (1+k 2 ) 1/2 (Japanese bracket). Notice that our definition is not exactly the same as that in [1,2]. In particular, the base space is T = R/(2πZ) in [1,2].…”

A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion

“…Eq. (1.2) was also studied by Barostichi, Himonas and Petronilho [19] and they exhibited a power series method in abstract Banach spaces equiped with analytic initial data, and established a Cauchy-Kovalevsky type theorem.…”

Decay properties of solutions to a 4-parameter family of wave equations

“…It was shown, however, that the solution map is Hölder continuous if H s is equipped with a weaker H r norm where r ∈ [0, s). The equation was also studied in Barostichi, Himonas and Petronilho in [1] where they exhibited a power series method in abstract Banach spaces provided analytic initial data, thereby establishing a Cauchy-Kovalevsky type theorem for the k − abc equation (1.1).…”