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Propagation of high peaks for the space-time fractional stochastic partial differential equations

Ngartelbaye Guerngar, Erkan Nane

Abstract

We study the space-time nonlinear fractional stochastic heat equation driven by a space-time white noise, \begin{align*} \partial_t^βu(t,x)=-(-Δ)^{α/2}u(t,x)+I_t^{1-β}\Big[σ(u(t,x))\dot{W}(t,x)\Big],\ \ t>0, \ x\in \mathbb{R} , \end{align*} where $σ:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function and the initial condition is a measure on $\mathbb{R}.$ Under some growth conditions on $σ,$ we derive two important properties about the moments of the solution: (i) For $p\geq 2,$ the $p^{\text{th}}$ absolute moment of the solution to the equation above grows exponentially with time. (ii) Moreover, the distances to the origin of the farthest high peaks of these moments grow exactly exponentially with time. Our results provide an extension of the work of Chen and Dalang (Stoch PDE: Anal Comp (2015) 3:360-397) to a time-fractional setting. We also show that condition (i) holds when we study the same equation for $x\in\mathbb{R}^d.$

Propagation of high peaks for the space-time fractional stochastic partial differential equations

Abstract

We study the space-time nonlinear fractional stochastic heat equation driven by a space-time white noise, \begin{align*} \partial_t^βu(t,x)=-(-Δ)^{α/2}u(t,x)+I_t^{1-β}\Big[σ(u(t,x))\dot{W}(t,x)\Big],\ \ t>0, \ x\in \mathbb{R} , \end{align*} where is a globally Lipschitz function and the initial condition is a measure on Under some growth conditions on we derive two important properties about the moments of the solution: (i) For the absolute moment of the solution to the equation above grows exponentially with time. (ii) Moreover, the distances to the origin of the farthest high peaks of these moments grow exactly exponentially with time. Our results provide an extension of the work of Chen and Dalang (Stoch PDE: Anal Comp (2015) 3:360-397) to a time-fractional setting. We also show that condition (i) holds when we study the same equation for
Paper Structure (6 sections, 16 theorems, 109 equations)

This paper contains 6 sections, 16 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Proofs of the main results
  5. Higher Dimension
  6. Appendix

Key Result

Theorem 2.1

Suppose $1<\alpha<2$ and $\sigma$ satisfies Eq. eq:upG with $\bar{\varsigma}=0.$ If for some $\eta>0,$ then there are positive constants $C<\infty$ and $b=\min(\eta, 2)$ such that for all $(t,x)\in[1,\infty)\times \mathbb{R}$, where $J_{\alpha,\beta}:=\overline{\mathcal{G}}_{\alpha,\beta}\star\mu$ and $\overline{\mathcal{G}}_{\alpha,\beta}$ is defined in RefKern.

Theorems & Definitions (31)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 21 more