Acta mathematica scientia, Series B >
REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION
Received date: 2012-04-05
Revised date: 2012-11-01
Online published: 2013-11-20
Supported by
Partially supported by projects: MNTR: 174024, and APV: 114-451-3605/2013.
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)), s ∈ R, to the nonlinear time fractional equation of distributed order with spatial Laplace operator subject to the Cauchy conditions
∫20p( β)D β*u(x, t)d = △xu(x, t) + f(t, u(t, x)), t ≥ 0, x ∈ Rn, 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., p( β) =∑mk=1bkδ( β−β 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), s ∈ R.
Mirjana STOJANOVIC . REGULARITY OF SOLUTIONS TO NONLINEAR TIME FRACTIONAL DIFFERENTIAL EQUATION[J]. Acta mathematica scientia, Series B, 2013 , 33(6) : 1721 -1735 . DOI: 10.1016/S0252-9602(13)60118-6
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