An L_q(L_p)-theory for the time fractional evolution equations with variable coefficients

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• 作者：Ildoo Kim^a ; ^{waldoo@kias.re.kr} ; Kyeong-Hun Kim^b ; ¹ ; ^{kyeonghun@korea.ac.kr} ; Sungbin Lim^b ; ^{sungbin@korea.ac.kr}
• 关键词：45D05 ; 45K05 ; 45N05 ; 35B65 ; 26A33
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：123-176
• 全文大小：704 K
• 卷排序：306

文摘

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.