Asymptotically short term behavior of solutions to one dimensional diffusion processes (Q1963971)

The authors study the short term behavior of solutions to diffusion processes: \[ \begin{alignedat}{2} &u_t = \frac{\partial}{\partial x}(p(x)u_x) + f(x,u,u_x), \qquad && 0<x<\pi, t>0,\\ &u(0,t) = u(\pi,t) = 0,\qquad && t>0,\\ &u(x,0) = u_0(x) \end{alignedat} \] in the case when \(u_0\) does not satisfy the homogeneous Dirichlet boundary conditions, \(p(x)\) is piecewise smooth, and \(f\) is continuous and Lipschitz in \(u\) and \(u_x\). The main result reads as: If \(u\) is a solution then \[ \lim_{t\to +0}\frac{1}{\sqrt t}\int_0^a(u_0(x) - u(x,t)) dx = \frac{2}{\sqrt\pi} \left(u_0(0)\sqrt{p(0)} + u_0(a)\sqrt{p(a)}\right). \]