{"title":"Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources","authors":"Dang Duc Trong, Nguyen Dang Minh, Nguyen Nhu Lan, Nguyen Thi Mong Ngoc","doi":"10.1007/s40306-023-00503-7","DOIUrl":null,"url":null,"abstract":"<div><p>We-24pt study the nonlinear stochastic time-fractional diffusion equation in the spatial domain <span>\\(\\mathbb {R}\\)</span> driven by a locally Lipschitz source satisfying </p><div><div><span>$$\\begin{aligned} \\left( {~}_{t}D_{0^{+}}^{\\alpha } - \\frac{\\partial ^{2} }{\\partial x^{2}}\\right) u(t,x) = I_{t}^{\\gamma }\\left( F(t,x,u)\\right) , \\end{aligned}$$</span></div></div><p>where <span>\\(x\\in \\mathbb {R},\\alpha \\in (0,1],\\gamma \\ge 1-\\alpha \\)</span>, the source term is defined <span>\\(F(t,x,u) = f(t,x,u(t,x))\\)</span> <span>\\( + \\rho (t,x,u(t,x))\\dot{W}(t,x)\\)</span> and <i>W</i> is the multiplicative space-time white noise. We investigate the existence, uniqueness of a maximal random field solution. Moreover, we prove the stability of the solution with respect to perturbed fractional orders <span>\\(\\alpha , \\gamma \\)</span> and the initial condition.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40306-023-00503-7.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-023-00503-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We-24pt study the nonlinear stochastic time-fractional diffusion equation in the spatial domain \(\mathbb {R}\) driven by a locally Lipschitz source satisfying
where \(x\in \mathbb {R},\alpha \in (0,1],\gamma \ge 1-\alpha \), the source term is defined \(F(t,x,u) = f(t,x,u(t,x))\)\( + \rho (t,x,u(t,x))\dot{W}(t,x)\) and W is the multiplicative space-time white noise. We investigate the existence, uniqueness of a maximal random field solution. Moreover, we prove the stability of the solution with respect to perturbed fractional orders \(\alpha , \gamma \) and the initial condition.
期刊介绍:
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.