  In this paper, we study a one dimensional nonlinear equation with diffusion
$-\nu(-\partial_{xx})^{\frac{\alpha}{2}}$ for $0\leq \alpha\leq 2$ and $\nu>0$.
We use a viscous-splitting algorithm to obtain global nonnegative weak
solutions in space $L^1(\mathbb{R})\cap H^{1/2}(\mathbb{R})$ when
$0\leq\alpha\leq 2$. For subcritical $1<\alpha\leq 2$ and critical case
$\alpha=1$, we obtain global existence and uniqueness of nonnegative spatial
analytic solutions. We use a fractional bootstrap method to improve the
regularity of mild solutions in Bessel potential spaces for subcritical case
$1<\alpha\leq 2$. Then, we show that the solutions are spatial analytic and can
be extended globally. For the critical case $\alpha=1$, if the initial data
$\rho_0$ satisfies $-\nu<\inf\rho_0<0$, we use the characteristics methods for
complex Burgers equation to obtain a unique spatial analytic solution to our
target equation in some bounded time interval. If $\rho_0\geq0$, the solution
exists globally and converges to steady state.
