On the Lyapunov functions for the solutions of the generalized Burgers equation (Q1424072)

The aim of this work is to prove that the generalized Burgers equation \[ \begin{cases} \partial_tu-\Delta u+\nabla u^{\alpha+1}= 0,\quad & x\in \mathbb{R}^d,\;t> 0,\\ u(0,x)= u_0(x),\quad & x\in\mathbb{R}^d\end{cases} \] has a family of Lyapunov functions not arising from the energy inequality. Here \(\alpha\) is an integer greater than 1 and the initial data \(u_0\) belongs to the homogeneous Besov spaces.