# Does the solution to $u_t=-uu_x+0.1u_{xx}$ decay in time?

Consider the following PDE: \begin{align} &u_t+uu_x=0.1u_{xx},\qquad 0<x<1,t>0\\ &u(x+1,t)=u(x,t),\qquad t\geq 0 \\ &u(x,0) = \sin 2\pi x,\qquad 0\leq x\leq 1 \end{align}

I used two different numerical schemes(Finite difference and Spectral method) and implemented it by MATLAB to plot $u$ at $t = 0.21$. The results are very different:

Here are my questions:

• Is there a name for this nonlinear PDE?
• Does the solution to the PDE decay in time?
• [EDITED:]Which one of the figures above is close to the true solution?
-

BTW, by decay you mean the $C^0$norm decay?