Fractional heat equations on a segment

Abstract

We study the initial-boundary-value problem for the fractional heat equation on a segment $ ( 0,a ) $ \begin{equation} \left \{ \begin{array}{c} u_{t}+\lambda \vert u \vert ^{\rho }\text{ }u+C_{\alpha }\partial _{x}^{\alpha }u=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=h_{1}(t),u_{x}(0,t)=h_{2}(t),t>0, \end{array} \right . \tag*{(0.1)} \end{equation} where $\lambda \in R$ , $\rho \geq 0$ , $\alpha \in ( 1,\frac{3}{2} ] $ $,$ the constant $C_{\alpha }$ is chosen by a dissipative condition, such that $\ \text{Re}C_{\alpha }p^{ [ \alpha ] +1- \{ \alpha \} }>0$ for $\text{Re}p=0$ and \begin{equation*} \partial _{x}^{\alpha }u=\int_{0}^{x}\frac{\partial _{s}^{ [ \alpha ] +1}u(s,t)}{ ( x-s ) ^{ \{ \alpha \} }}ds. \end{equation*} Here $ [ \alpha ] $ and $ \{ \alpha \} $ are integer and fractional parts of $\alpha .$ The aim of this paper is to prove the global existence of solutions to the initial-boundary-value problem (0.1) and to find the main term of the asymptotic representation of solutions.