Local Well-posedness of Nonlinear Time–fractional Diffusion Equation: Apassara Suechoei, Parinya Sa Ngiamsunthorn

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Local Well-posedness of Nonlinear Time–fractional Diffusion Equation

Apassara Suechoei, Parinya Sa Ngiamsunthorn

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Abstract

We study the local well-posedness of the following time-fractional nonlinear diffusion equation

$\begin{align*} \begin{cases} ^CD_{0,t}^{\alpha ,\lambda }u - \Delta u = {\text{ }}{\left| u \right|^{p - 1}}u,\quad x \in {\mathbb{R}^n},\,\,t > 0 \\ u(x,0) = {u_0}(x), \quad x \in \mathbb{R}^{n} \end{cases} \end{align*}$

where $0 < \alpha < 1,\lambda \geqslant 0, p>1 , {u_0} \in {C_0}({\mathbb{R}^n})$ and $^CD_{0,t}^{\alpha ,\lambda }$ denotes Caputo tempered fractional derivative of order $\alpha$ . The local existence and uniqueness results are obtained from heat kernel and fixed point theorem. Then, we extend the solution to establish a maximal mild solution. Moreover, we provide estimate for continuous dependence on initial condition.

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Published

2021-09-01

How to Cite

Team, S. (2021). Local Well-posedness of Nonlinear Time–fractional Diffusion Equation: Apassara Suechoei, Parinya Sa Ngiamsunthorn. Thai Journal of Mathematics, 19(3), 865–884. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1202

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Vol. 19 No. 3 (2021): September

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Articles