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Non-linear Noise Excitation for some Space-Time Fractional Stochastic Equations in Bounded Domains

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Abstract

In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains:

$$\partial^\beta_tu_t(x) = -\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u) \stackrel{\cdot}{F}(t,x)] $$

in (d+1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator \(\partial^\beta_t\) is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process and I1−βt is the fractional integral operator. The forcing noise denoted by \(\stackrel{\cdot}{F}(t,x)\) is a Gaussian noise. The multiplicative non-linearity σ: ℝ → ℝ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane [32]. We first study the existence and uniqueness of the solution of these equations and under suitable conditions on the initial function, we also study the asymptotic behavior of the solution with respect to the parameter λ. In particular, our results are significant extensions of those in [14], [16], [32], and [33].

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK

    Mohammud Foondun

  2. Department of Mathematics, Georgia College & State University, 231 W. Hancock St, Milledgeville, GA, 31061, USA

    Jebessa B. Mijena

  3. Department of Mathematics and Statistics, Auburn University 221 Parker Hall, Auburn, AL, 36830, USA

    Erkan Nane

Authors
  1. Mohammud Foondun
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  2. Jebessa B. Mijena
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  3. Erkan Nane
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Correspondence to Mohammud Foondun.

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Foondun, M., Mijena, J.B. & Nane, E. Non-linear Noise Excitation for some Space-Time Fractional Stochastic Equations in Bounded Domains. FCAA 19, 1527–1553 (2016). https://doi.org/10.1515/fca-2016-0079

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