### Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative

#### 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 $L^q(0, T; L^p(\mathbf R^n))$.

#### Article information

Source
Adv. Differential Equations Volume 7, Number 2 (2002), 217-236.

Dates
First available in Project Euclid: 27 December 2012