The Cauchy problem for evolution equations with the Bessel operator of infinite order. I

2010 ◽  
Vol 54 (6) ◽  
pp. 1-12 ◽  
Author(s):  
V. V. Gorodetskii ◽  
O. V. Martynyuk
Keyword(s):  
Cauchy Problem ◽  
Infinite Order ◽  
Evolution Equations ◽  
Bessel Operator ◽  
The Cauchy Problem

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.

A note on optimal L 2 -estimates of solutions to some strongly damped σ-evolution equations

2020 ◽  
Vol 121 (1) ◽  
pp. 59-74
Author(s):  
Ryo Ikehata
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Evolution Equations ◽  
Estimates Of Solutions ◽  
The Cauchy Problem ◽  
Strongly Damped

We consider the Cauchy problem in R n for the so-called σ-evolution equations with damping terms. We derive asymptotic profiles of solutions with weighted L 1 , 1 ( R n ) initial data, and investigates the optimality of estimates of solutions in L 2 -sense. The obtained results will generalize and compensate those already known in (J. Math. Anal. Appl. 478 (2019) 476–498, J. Diff. Eqns 257 (2014) 2159–2177, Diff. Int. Eqns 30 (2017), 505–520).

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