数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (6): 1721-1735.doi: 10.1016/S0252-9602(13)60118-6

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REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

Mirjana STOJANOVIC   

  1. Department of Mathematics and Informatics, University of Novi Sad, Trg D.Obradovi´ca 4, 21 000 Novi Sad, Serbia
  • 收稿日期:2012-04-05 修回日期:2012-11-01 出版日期:2013-11-20 发布日期:2013-11-20
  • 基金资助:

    Partially supported by projects: MNTR: 174024, and APV: 114-451-3605/2013.

REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION

Mirjana STOJANOVIC   

  1. Department of Mathematics and Informatics, University of Novi Sad, Trg D.Obradovi´ca 4, 21 000 Novi Sad, Serbia
  • Received:2012-04-05 Revised:2012-11-01 Online:2013-11-20 Published:2013-11-20
  • Supported by:

    Partially supported by projects: MNTR: 174024, and APV: 114-451-3605/2013.

摘要:

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C(t ∈ (0,∞); Hs+22 (Rn)) ∩ C0(t ∈ [0,∞); Hs(Rn)), sR, to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions

20pβ)β*u(x, t)d = △xu(x, t) + f(t, u(t, x)), t ≥ 0, xRn, u(0, x) = φ(x), ut(0, x) = Ψ (x), (0.1)

where △x is 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., pβ) =∑mk=1bkδββ k), 0 < k < 2, bk > 0,
k = 1, 2, · · · , m.
The regularity of the solution is established in the framework of the space C(t ∈ (0, ∞); C(Rn)) ∩C0(t ∈ [0,∞); C∞(Rn)) when the initial data belong to the Sobolev space Hs2(Rn), sR.

关键词: nonlinear time-fractional equations of distributed order, existence-uniqueness theorems, viscosity solutions, regularity result

Abstract:

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C(t ∈ (0,∞); Hs+22 (Rn)) ∩ C0(t ∈ [0,∞); Hs(Rn)), sR, to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions

20pβ)β*u(x, t)d = △xu(x, t) + f(t, u(t, x)), t ≥ 0, xRn, u(0, x) = φ(x), ut(0, x) = Ψ (x), (0.1)

where △x is 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., pβ) =∑mk=1bkδββ k), 0 < k < 2, bk > 0,
k = 1, 2, · · · , m.
The regularity of the solution is established in the framework of the space C(t ∈ (0, ∞); C(Rn)) ∩C0(t ∈ [0,∞); C∞(Rn)) when the initial data belong to the Sobolev space Hs2(Rn), sR.

Key words: nonlinear time-fractional equations of distributed order, existence-uniqueness theorems, viscosity solutions, regularity result

中图分类号: 

  • 26A33