Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4665011 | Advances in Mathematics | 2017 | 54 Pages |
Abstract
We introduce an Lq(Lp)Lq(Lp)-theory for the semilinear fractional equations of the typeequation(0.1)∂tαu(t,x)=aij(t,x)uxixj(t,x)+f(t,x,u),t>0,x∈Rd. Here, α∈(0,2)α∈(0,2), p,q>1p,q>1, and ∂tα is the Caupto fractional derivative of order α . Uniqueness, existence, and Lq(Lp)Lq(Lp)-estimates of solutions are obtained. The leading coefficients aij(t,x)aij(t,x) are assumed to be piecewise continuous in t and uniformly continuous in x . In particular aij(t,x)aij(t,x) are allowed to be discontinuous with respect to the time variable. Our approach is based on classical tools in PDE theories such as the Marcinkiewicz interpolation theorem, the Calderon–Zygmund theorem, and perturbation arguments.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Ildoo Kim, Kyeong-Hun Kim, Sungbin Lim,