On the uniqueness of solutions to parabolic semilinear problems under nonlocal conditions with integrals

The uniqueness of classical solutions to parabolic semilinear problems together with nonlocal initial conditions with integrals, for the operator ∑i,j=1n∂∂xi(aij(x,t)∂∂xj)+c(x,t)-∂∂t,\sum\limits_{i,j = 1}^n {{\partial \over {\partial {x_i}}}\left( {{a_{ij}}\left( {x,t} \right){\partial \over {\partial {x_j}}}} \right)} + c\left( {x,t} \right) - {\partial \over {\partial t}},, x=(x₁,...,x_n), in the cylindrical domain D:D₀ ×(t₀, t₀+T) ⊂ ℜⁿ⁺¹, where t₀ ∈ ℜ, 0 < T < ∞, are studied. The result requires that the nonlocal conditions with integrals be introduced.