Abstract
In this paper we study a class of nonlinear integro-differential equations which correspond to a fractional-order time derivative and interpolate nonlinear heat and wave equations. For this purpose we first establish some space--time estimates of the linear flow which is produced by Mittag--Leffler's functions based on Mihlin--Hörmander's multiplier estimates and other harmonic analysis tools. Using these space--time estimates we prove the well-posedness of a local mild solution of the Cauchy problem for the nonlinear integro-differential equation in $ C([0,T); L^p(\mathbf R^n))$ or $L^q(0, T; L^p(\mathbf R^n))$.
Citation
Hitoshi Hirata. Changxing Miao. "Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative." Adv. Differential Equations 7 (2) 217 - 236, 2002. https://doi.org/10.57262/ade/1356651852
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