The existence and uniqueness of a classical solution to the Cauchy problem for a nonlinear parabolic equation of the second order with coefficients of the linear part increasing as $|x|\to \infty$ are established. This solution belongs to the class of H?lder functions with $t\in(0;T_0]$, where $T_0\in (0,T]$ is sufficiently small. With the help of this result, the existence and uniqueness of a solution to the problem of determining the coefficient of an unknown function in the corresponding linear parabolic equation with coefficients independent of $t$ are proved.

[1] Eidelman S. D. Parabolic systems. North-Holland, Amsterdam, 1969.
[2] Medynsky I.P. Eidelman’s study of nonlinrar problems and their development. Nauk. Visnyk Cherniv. Univer. 2011. 1. (1–2). 114–128. (in Ukrainian)
[3] Beznoshchenko N.Ya. On the Cauchy problem for the equation ut −Δu+u Au = f. Diff. Eq. 1983. 19, (6), 991–1000.
[4] Lavrenchuk V. P. Cauchy problem for a nonlinear secon-jrder parabolic equation with unlimited coefficient and degenerate. Nauk. Visnyk Cherniv. Univer. 2000. (76). 50–53. (in Ukrainian)
[5] Aronson D.G., Besala P. Parabolic equations with unbounded coefficients. J. of Diff. Eq. 1967. 3 (1), 1–14.
[6] Ivasyshen S.D., Lavrenchuk V.P., Pasichnyk H.S. On the Cauchy problem for a quasilinear parabolic equation with increasing coefficients. Diff. Eq. and their applications: Proceedings of the Intern. Scientific Conf. dedicated to the 70th anniversary of Prof. V.V. Marints. Uzhgorod, 2012. 36. (in Ukrainian)