scholarly journals The cauchy problem for non-linear Klein-Gordon equations

1993 ◽  
Vol 152 (3) ◽  
pp. 433-478 ◽  
Author(s):  
Jacques C. H. Simon ◽  
Erik Taflin
Keyword(s):  
Cauchy Problem ◽  
Non Linear ◽  
The Cauchy Problem ◽  
Klein Gordon

scholarly journals Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions

2020 ◽  
Vol 23 (6) ◽  
pp. 1663-1677
Author(s):  
Michael Ruzhansky ◽  
Berikbol T. Torebek
Keyword(s):  
Cauchy Problem ◽  
Exponential Function ◽  
Evolution Equations ◽  
Oscillatory Integrals ◽  
Schrödinger Equations ◽  
Fractional Evolution Equations ◽  
Type Function ◽  
Fractional Schrödinger Equations ◽  
The Cauchy Problem ◽  
Klein Gordon

Abstract The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E α, β (i λ ϕ(x)), x ∈ ℝ N and E α, β (i α λ ϕ(x)), x ∈ ℝ N for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

scholarly journals Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms

2015 ◽  
Vol 35 (10) ◽  
pp. 4889-4903 ◽  
Author(s):  
Michinori Ishiwata ◽  
◽  
Makoto Nakamura ◽  
Hidemitsu Wadade ◽  
◽  
...  
Keyword(s):  
Cauchy Problem ◽  
The Cauchy Problem ◽  
Klein Gordon
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