Blow-up and global existence of solutions for time-space fractional pseudo-parabolic equation

Abstract

<abstract><p>In this article, we consider the Cauchy problem for the following time-space fractional pseudo-parabolic equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{l} { }_{0}^{C} D_{t}^{\alpha}(I-m \Delta ) u+\left ( - \Delta \right ) ^{\frac{\beta }{2} } u = |u|^{p-1} u, \quad x \in \mathbb{R}^{N}, \quad t&gt;0, \\ u(0, x) = u_{0}(x), \quad\quad\quad\quad\quad\quad\quad\qquad x \in \mathbb{R}^{N}, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ 0 &lt; \alpha &lt; 1, \ 0 &lt; \beta &lt; 2, \ p &gt; 1, \ m &gt; 0, \ u_{0} \in L^{q}\left(\mathbb{R}^{N}\right) $. An estimating $ L^p-L^q $ for solution operator of time-space fractional pseudo-parabolic equations is obtained. The critical exponents of this problem are determined when $ u_0\in L^{q}(\mathbb{R}^{N}). $ Moreover, we also obtain global existence of the mild solution when $ u_0\in L^p(\mathbb{R}^{N})\cap L^q(\mathbb{R}^{N}) $ small enough.</p></abstract>