In this paper, we investigate initial boundary value problems for the following time-space fractional Rosenau equationwith the Caputo time fractional derivatives and the spectral fractional Laplacian operators. To the best of our knowledge, there are few results concerning the time-space fractional Rosenau equations. The main difficulties are the nonlocal effects generated by the operators 𝜕 𝛽 t and (− Δ) 𝛾 . First, we establish some rough and rigorous decay estimates of weak solutions to the corresponding linear equations, respectively. Based on the decay estimates, under small initial value condition, we prove the global existence and asymptotic behavior of weak solutions in the time-weighted Sobolev spaces by the contraction mapping principle. Furthermore, we discuss the regularity of weak solutions when initial value data are strengthened from H s (Ω) to H s+1 (Ω).
<abstract><p>In this paper, we study initial boundary value problems for the following fully nonlocal Boussinesq equation</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ _0^{C}D_{t}^{\beta}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma}{_0^{C}D_{t}^{\beta}}u = -(-\Delta)^{\sigma}f(u) $\end{document} </tex-math></disp-formula></p>
<p>with spectral fractional Laplacian operators and Caputo fractional derivatives. To our knowledge, there are few results on fully nonlocal Boussinesq equations. The main difficulty is that each term of this equation has nonlocal effect. First, we obtain explicit expressions and some rigorous estimates of the Green operators for the corresponding linear equation. Further, we get global existence and some decay estimates of weak solutions. Second, we establish new chain and Leibnitz rules concerning $ (-\Delta)^{\sigma} $. Based on these results and small initial conditions, we obtain global existence and long-time behavior of weak solutions under different dimensions $ N $ by Banach fixed point theorem.</p></abstract>
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