Asymptotic behaviour of a generalized Burgers' equation (Q1305133)

Nonnegative solutions of the equation \[ u_t=\Delta (u^m) - {\partial\over{\partial x_1}}(u^q) , \qquad x\in \mathbb R^N,\quad m>1,\quad q=m+\tfrac 1N , \] are studied. It is shown that their behavior as \(t\to\infty\) is given by a one-parametric family of self-similar solutions.