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,∞);H2s+2(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 equation(0.1)∫02p(β)D*β u(x,t)dβ=Δxu(x,t)+f(t,u(t, x)),t≥0,x∈Rn,u(o,x)=ut(0,x)=ψ(x), where Δx 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)=∑k=1mbk δ(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∞(Rn))∩Co(t∈[0,∞);C∞(Rn))C∞(t∈(0,∞); C∞(Rn))∩Co(t∈[0,∞);C∞(Rn)) when the initial data belong to the Sobolev space H2s(Rn), s ∈ R.