Decay rates for second-order linear evolution problems with fractional laplacian operators

Authors

DOI:

https://doi.org/10.5902/2179460X41963

Keywords:

Asymptotic behavior, fractional Laplace operator, Fourier space, second-order equations.

Abstract

In this work we study the asymptotic behavior of solutions for a general linear second-order evolution differential equation in time with fractional Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with less demand on the initial data when compared to previous results in the literature. In certain cases, we observe that the dissipative structure of the equation is of regularity-loss type. Due to that special structure, to get decay estimates in high frequency region in the Fourier space it is necessary to impose additional regularity on the initial data to obtain the same decay estimates as in low frequency region. The results obtained in this work can be applied to several initial value problems associated to second-order equations, as for example, wave equation, plate equation, IBq, among others. 

Author Biographies

Cleverson Roberto da Luz, Universidade Federal de Santa Catarina, Florianópolis, SC

Doutor em Matemática pela Universidade Federal do Rio de Janeiro. Atualmente é professor Associado II e coordenador da Câmara de Pesquisa do Departamento de Matemática na Universidade Federal de Santa Catarina.

Maíra Fernandes Gauer Palma, Universidade Federal de Santa Catarina, Florianópolis, SC

Possui graduação em Matemática (Licenciatura) pela Universidade Federal de Santa Catarina, mestrado e doutorado em Matemática e Computação Científica pela Universidade Federal de Santa Catarina. Atualmente é professora adjunta do Departamento de Engenharias da Mobilidade na Universidade Federal de Santa Catarina.

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Published

2021-03-01

How to Cite

Luz, C. R. da, & Palma, M. F. G. (2021). Decay rates for second-order linear evolution problems with fractional laplacian operators. Ciência E Natura, 43, e14. https://doi.org/10.5902/2179460X41963