Global existence, asymptotic behavior, and regularity of solutions for time–space fractional Rosenau equations

Abstract

In this paper, we investigate initial boundary value problems for the following time–space fractional Rosenau equation with 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 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 to .