The well-posedness of semilinear fractional dissipative equations on [formula omitted]

Abstract

In this paper, we study the well-posedness of time-space fractional dissipative equations. Combining the Hörmander-multipliers theory and asymptotic property of the Mittag-Leffler functions, we give a useful method to estimate the solution operator which is independent of the C0-semigroup generated by the fractional Laplace operator (−Δ)β2 and the Mainardi's Wright type function. Furthermore, we discuss the global/local well-posedness in some appropriate time-space Lebesgue space and Besov spaces. We also obtain several results about blow-up alternative and asymptotic behavior. The approaches are based on the Hörmander-multipliers theory, Gagliardo-Nirenberg inequalities, fixed point techniques, Sobolev embedding and some methods in harmonic analysis.