Heat equation with degeneration in Holder and Slobodetskii spaces

We study initial-boundary value problems for the heat equation in which heat conductivity $\alpha^2(x)$ may depend on the space variable $x\in\mathbb R^+$; the nonnegative function $\alpha(x)$ is allowed to tend to infinity (respectively, zero) as $x\to+\infty$ (respectively, $x\to+0$). We prove that these problems are well posed and examine the smoothness of solutions. It is shown that criteria for smoothness of the solutions can be stated in terms of certain functionals, namely, the Hölder constant (for Hölder spaces) and the generalized Hölder constant (for Slobodetskii spaces).