The Cauchy problem for the heat equation with a strongly variable coefficient (Q1594324)

The authors consider an equation of the form: \[ \frac{\partial u}{\partial t}- \alpha (x) \frac{\partial}{\partial x} (\alpha (x) \frac{\partial u}{\partial x})=f(x,t),\quad x\in \mathbb{R},\;t>0. \] The smoothness of the solution of this equation is studied under the assumption that the coefficient \(\alpha\) is positive, has an integrable singularity at \(x=0\) and is nonintegrable as \(x \rightarrow \infty.\)