Asymptotic behavior of the solutions of a nonlinear Fokker--Planck equation with Dirichlet boundary conditions (Q1260839)

The convergence of the solution \(u(x,t)\) of the problem \[ u_ t=(u^ m)_{xx}+(u^ n)_{x'},\;x>0,t>0, \] \[ u(x,0)=u_ 0(x),\;x>0,\quad u(0,t)=\psi(t),\;t>0 \] to the corresponding stationary solution under suitable assumptions on data \(u(0,t)\) and \(u(x,0)\) is proved. For the proof the authors construct a family of travelling wave solutions, depending on a parameter \(h\in\mathbb{N}\), in such a way that each of them is a supersolution up to a given time \(t_ h\) \((t_ h\to\infty\) as \(h\to\infty)\). Then the convergence to the stationary solution in the cases \(n<1\), and \(m>n>1\) is considered.