On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise

Hoang The Tuan

doi:10.3934/dcdsb.2020318

Article Contents

2021, Volume 26, Issue 3: 1749-1762. Doi: 10.3934/dcdsb.2020318

This issue Previous Article Existence results for fractional differential equations in presence of upper and lower solutions Next Article

On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise

Hoang The Tuan^,

Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam

Received: February 2020

Revised: July 2020

Early access: November 2020

Published: March 2021

This research is supported by a grant from the Vietnam Academy of Science and Technology under the grant number DLTE00.01–20/21.

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Abstract

This paper is devoted to study of time-fractional elliptic equations driven by a multiplicative noise. By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional differential equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov–Perron operator defined from this representation of solutions, we show the asymptotic behaviour of solutions to these systems in the mean square sense. As a consequence, we also prove existence, uniqueness and the convergence rate of their solutions.

Keywords:

Stochastic partial differential equations,
time fractional derivatives,
mild solution,
existence and uniqueness,
asymptotic behavior.

Mathematics Subject Classification: Primary: 34A08, 35A01, 35B20; Secondary: 35B40, 60H15, 35R60.

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