Advances in Differential Equations

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

Hitoshi Hirata and Changxing Miao

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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

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651852

Mathematical Reviews number (MathSciNet)
MR1869562

Zentralblatt MATH identifier
1032.45009

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35R10: Partial functional-differential equations 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]

Citation

Hirata, Hitoshi; Miao, Changxing. Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative. Adv. Differential Equations 7 (2002), no. 2, 217--236. https://projecteuclid.org/euclid.ade/1356651852.


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