Local Well-posedness of Nonlinear Time–fractional Diffusion Equation
Apassara Suechoei, Parinya Sa Ngiamsunthorn
Abstract
We study the local well-posedness of the following time-fractional nonlinear diffusion equation
&space;=&space;{u_0}(x),&space;\quad&space;x&space;\in&space;\mathbb{R}^{n}&space;\end{cases}&space;\end{align*} "\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 
denotes Caputo tempered fractional derivative of order 
. 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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|ISSN 1686-0209|
