REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）...

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Bibliographic Details
Published inActa mathematica scientia Vol. 33; no. 6; pp. 1721 - 1735
Main Author STOJANOVIC, Mirjana
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.11.2013
Department of Mathematics and Informatics, University of Novi Sad, Trg D.0bradovi′ca 4, 21 000 Novi Sad, Serbia
Subjects
26A33
35B40
35B65
45K05
45L05
Differential equations
existence-uniqueness theorems
Mathematical analysis
nonlinear time-fractional equations of distributed order
Nonlinearity
Operators
Proving
Regularity
regularity result
Sobolev空间
S空间
Viscosity
viscosity solutions
三角形分布
分数阶微分方程
存在唯一性
拉普拉斯算子
柯西分布
非线性时间
regularity result
45L05
26A33
35B65
45K05
existence-uniqueness theorems
viscosity solutions
35B40
nonlinear time-fractional equations of distributed order
Online AccessGet full text
ISSN0252-9602
1572-9087
DOI10.1016/S0252-9602(13)60118-6

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Summary:We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）＋f（t,u（t,x））,t≥0,x∈R^n,u（0,x）=φ（x）,ut（0,x）=ψ（x）,（0.1） where △xis the spatial Laplace operator,D＊^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval （0, 2）, i.e., p（β） =m∑k=1bkδ（β-βk）,0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞（t∈（0,∞）;C^∞（R^n））∩C^0（t∈[0,∞）;C^∞（R^n））when the initial data belong to the Sobolev space H2^8（R^n）,s∈R.
Bibliography:42-1227/O
nonlinear time-fractional equations of distributed order; existence-uniqueness theorems; viscosity solutions; regularity result
We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞（t∈（0,∞）;H2^s＋2（R^n））∩C^0（t∈[0,∞）;H^s（R^n））,s∈R,to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions ∫0^2p（β）D＊^βu（x,t）dβ=△xu（x,t）＋f（t,u（t,x））,t≥0,x∈R^n,u（0,x）=φ（x）,ut（0,x）=ψ（x）,（0.1） where △xis the spatial Laplace operator,D＊^β is the operator of fractional differentiation in the Caputo sense and the force term F satisfies the Assumption 1 on the regularity and growth. For the weight function we take a positive-linear combination of delta distributions concentrated at points of interval （0, 2）, i.e., p（β） =m∑k=1bkδ（β-βk）,0〈βk〈2,bk〉0,k=1,2,…,m.The regularity of the solution is established in the framework of the space C^∞（t∈（0,∞）;C^∞（R^n））∩C^0（t∈[0,∞）;C^∞（R^n））when the initial data belong to the Sobolev space H2^8（R^n）,s∈R.
Mirjana STOJANO VIC（Department of Mathematics and Informatics, University of Novi Sad, Trg D. ObradoviSa 4, 21 000 Novi Sad, Serbia）
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content type line 23
ISSN:0252-9602
1572-9087
DOI:10.1016/S0252-9602(13)60118-6