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Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources

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Abstract

We-24pt study the nonlinear stochastic time-fractional diffusion equation in the spatial domain \(\mathbb {R}\) driven by a locally Lipschitz source satisfying

$$\begin{aligned} \left( {~}_{t}D_{0^{+}}^{\alpha } - \frac{\partial ^{2} }{\partial x^{2}}\right) u(t,x) = I_{t}^{\gamma }\left( F(t,x,u)\right) , \end{aligned}$$

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.

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Acknowledgements

The authors thank an anonymous referee and the editor for helpful comments that improved the quality and presentation of the paper. The research was supported by Vietnam National University of Hochiminh City [Grant No. B2021-18-02].

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science of Ho Chi Minh City, No. 227 Nguyen Van Cu Street, Ward 4, District 5, Ho Chi Minh City, Vietnam

    Dang Duc Trong & Nguyen Thi Mong Ngoc

  2. Vietnam National University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Dang Duc Trong & Nguyen Thi Mong Ngoc

  3. Department of Fundamental Studies, Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam

    Nguyen Dang Minh

  4. Department of Economic Mathematics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Nhu Lan

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  1. Dang Duc Trong
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  2. Nguyen Dang Minh
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  3. Nguyen Nhu Lan
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  4. Nguyen Thi Mong Ngoc
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Correspondence to Nguyen Dang Minh.

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Dedicated to Professor Duong Minh Duc on the occasion of his 70th birthday.

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Trong, D.D., Minh, N.D., Lan, N.N. et al. Continuity of the Solution to a Stochastic Time-fractional Diffusion Equations in the Spatial Domain with Locally Lipschitz Sources. Acta Math Vietnam 48, 237–257 (2023). https://doi.org/10.1007/s40306-023-00503-7

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