scholarly journals On the Convergence of Solutions for SPDEs under Perturbation of the Domain

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Zhongkai Guo ◽  
Jicheng Liu ◽  
Wenya Wang
Keyword(s):  
Boundary Condition ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Dirichlet Boundary Condition ◽  
Stochastic Partial Differential Equations ◽  
Dirichlet Boundary ◽  
Mild Solutions ◽  
Partial Differential ◽  
Domain Perturbation ◽  
Convergence Of Solutions

We investigate the effect of domain perturbation on the behavior of mild solutions for a class of semilinear stochastic partial differential equations subject to the Dirichlet boundary condition. Under some assumptions, we obtain an estimate for the mild solutions under changes of the domain.

scholarly journals Space-time fractional stochastic partial differential equations with Lévy noise

2020 ◽  
Vol 23 (1) ◽  
pp. 224-249
Author(s):  
Xiangqian Meng ◽  
Erkan Nane
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Linear Time ◽  
Mild Solutions ◽  
Stable Process ◽  
Random Measure ◽  
Type Equation ◽  
Poisson Random Measure ◽  
Partial Differential

AbstractWe consider non-linear time-fractional stochastic heat type equation$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg] \end{array} $$and$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h)\stackrel{\cdot}{N }(t,x,h)\bigg] \end{array} $$in (d + 1) dimensions, where α ∈ (0, 2] and d < min{2, β−1}α, ν > 0, $\begin{array}{} \partial^\beta_t \end{array} $ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, $\begin{array}{} I^{1-\beta}_t \end{array} $ is the fractional integral operator, N(t, x) are Poisson random measure with Ñ(t, x) being the compensated Poisson random measure. σ : ℝ → ℝ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in [16, 33]. Under the linear growth of σ, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when σ grows faster than linear.

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