Existence of classical solutions for singular parabolic problems

Abstract

Let L u ≡ u x x + b u x / x − u t Lu \equiv {u_{xx}} + b{u_x}/x - {u_t} with b b a constant less than 1. Its Green’s function corresponding to first boundary conditions is constructed by eigenfunction expansion. With this, a representation formula is established to obtain existence of a classical solution for the linear first initial-boundary value problem. Uniqueness of a solution follows from the strong maximum principle. Properties of Green’s function and of the solution are also investigated.