Abstract:

We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C^∞(t ∈ (0,∞); H^s+2₂ (Rⁿ)) ∩ C⁰(t ∈ [0,∞); H^s(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), u_t(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) =∑^m_k=1b_kδ( β−β _k), 0 < k < 2, b_k > 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 H^s₂(Rⁿ), s ∈ R.

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