On a viscous Hamilton–Jacobi equation with an unbounded potential term

We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton–Jacobi or eikonal equation. The equation includes an unbounded potential term V(x)V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf–Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω2|x|2/2V(x)=ω2|x|2/2.