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Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise

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Abstract

In this paper, we study the intermittent property for the following nonlinear stochastic partial differential equation (SPDE in the sequel) in (1+1)-dimension

$$\begin{aligned} \left( \frac{\partial }{\partial t}+q(x,D_x)\right) u(t,x)= g(u(t,x))\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x),\quad t>0 \quad \mathrm{and} \quad x\in {\mathbb {R}}, \end{aligned}$$

with \(q(x,D_x)\) is a pseudo-differential operator which generates a stable-like process. The forcing noise denoted by \(\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x)\) is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet \(w_\rho (t,x)\), we prove that the solution is weakly full intermittent based on the moment estimates of the solution.

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  1. Department of Statistics, Nanjing Audit University, Nanjing, People’s Republic of China

    Junfeng Liu

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Partially supported by National Natural Science Foundation of China (No. 11771209), Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 18YJCZH101), Natural Science Foundation of Jiangsu Province of China (No. BK20161579), Major Research Plan of Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 18KJA110002), QingLan Project and 333 Talent Training Project of Jiangsu Province (No. BRA2018357).

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Liu, J. Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise. Nonlinear Differ. Equ. Appl. 26, 1 (2019). https://doi.org/10.1007/s00030-018-0548-0

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