We find an upper viscosity solution and give a proof of the existence-uniqueness in the space C ∞ ( t ∈ ( 0 , ∞ ) ; H 2 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 (0.1) ∫ 0 2 p ( β ) D * β u ( x , t ) d β = Δ x u ( x , t ) + f ( t , u ( t , x ) ) , t ≥ 0 , x ∈ R n , u ( o , x ) = u t ( 0 , x ) = ψ ( x ) , where Δ 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 ) = ∑ k = 1 m b k δ ( β - β k ) , 0 < β 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 ∞ ( R n ) ) ∩ C o ( t ∈ [ 0 , ∞ ) ; C ∞ ( R n ) ) when the initial data belong to the Sobolev space H 2 s ( R n ) , s ∈ R .