On some degenerate differential operators on weighted function spaces (Q1378391)

The initial-boundary value problem \[ {\partial u\over\partial t} (x, t)= \alpha(x) {\partial^2u\over\partial x^2} (x, t),\quad x>0,\quad t\geq 0,\quad u(x,0)= u_0(x),\quad x\geq 0, \] \[ \lim_{x\to 0^+} \alpha(x) {\partial^2u\over\partial x^2} (x, t)= \lim_{x\to\infty} {\alpha(x)\over 1+x^2} {\partial^2u\over\partial x^2} (x,t)= 0,\quad t\geq 0, \] is studied. The initial datum \(u_0\) belongs to a suitable subspace of the Banach space of all continuous functions \(f: [0,+\infty)\to \mathbb{R}\) such that \(\lim_{x\to\infty} f(x)/(1+ x^2)= 0\). The function \(\alpha\) is continuous and positive on \([0,+\infty)\) and differentiable at \(0\). The existence and uniqueness of solutions is given by using semigroup techniques. Also the approximation of the solution by means of discrete-type positive operators is given.