On Wiman-Valiron type estimations for parabolic equations

In the paper we established Wiman-Valiron type estimations for parabolic equations of the form in Hilbert space, where A(t) is a uniformly positive-definite self-adjoint operator with a discrete spectrum. Imposing asymptotic character conditions on the distribution function N() of the eigen-values of the operator A(t), we derive estimations for the norm of the solution of the equation that in particular characterize the behavior of the solution as t ! 0 depending on the behavior of the Fourier coefficients of initial data.