Consider the initial value problem:
$u_t-(xu)_x = 0$, $\qquad$ $u(x,0) = \left\{\begin{array}{ll}\cos^2(\pi x/2), & -1\leq x \leq 1,\\0,&\text{otherwise}. \end{array}\right.$
I have tried separation of variables, but I am stuck on the eigenvalue problem. Assume $u(x,t) = X(x)T(t)$, then$$\begin{array}{l} X(x)T'(t)-(xX(x)T(t))_x = 0,\\ X(x)T'(t)-T(t)[X(x)+xX'(x)] = 0. \end{array}$$ Thus assume $$\frac{T'(t)}{T(t)} = \frac{X(x) + xX'(x)}{X(x)} = -\lambda.$$ Hence we have $$\left\{\begin{array}{ll} xX'(x) + (1+\lambda)X(x) = 0, &-1\leq x\leq 1,\\ T'(t)+\lambda T(t) =0, & t>0,\\ X(-1) = 0, \quad X(1)=0. \end{array}\right. $$
Whether there is an analytic solution for this PDE? How to calculate it if there exists an analytic solution? Any hints or bibliography will be appreciated.