Limit passage in quasilinear parabolic equations with weakly converging coefficients, and the asymptotic behavior of solutions of the Cauchy problem

Asymptotic closeness as $t\to+\infty$ (for each $x\in R^n$) is proved for solutions of two distinct Cauchy problems for quasilinear parabolic equations under the condition that certain limit means of the difference of the coefficients and of the difference of the initial functions are equal to zero. This proof is based on reducing the initial problem to a problem on the passage to the limit in a sequence of equations with weakly converging coefficients which is also of independent interest.