Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains

Abstract

For a fixed bounded domain D ⊂ R N we investigate the asymptotic behaviour for large times of solutions to the p-Laplacian diffusion equation posed in a tubular domainwith p > 2, i.e., the slow diffusion case, and homogeneous Dirichlet boundary conditions on the tube boundary. Passing to suitable re-scaled variables, we show the existence of a travelling wave solution in logarithmic time that connects the level u = 0 and the unique nonnegative steady state associated to the re-scaled problem posed in a lower dimension, i.e. in D ⊂ R N .We then employ this special wave to show that a wide class of solutions converge to the universal stationary profile in the middle of the tube and at the same time they spread in both axial tube directions, miming the behaviour of the travelling wave (and its reflection) for large times.The first main feature of our analysis is that wave fronts are constructed through a (nonstandard) combination of diffusion and absorbing boundary conditions, which gives rise to a sort of Fisher-KPP long-time behaviour. The second one is that the nonlinear diffusion term plays a crucial role in our analysis. Actually, in the linear diffusion framework p = 2 solutions behave quite differently.

Convergence in relative error for the Porous Medium equation in a tube

“…Step 3. On the other hand, we have 6 The speed c * is a monotone and continuous function of the domain.…”

“…Even though the tubular setting is intermediate between the above two and presents significant novelties in the description of the long-time behaviour, it has been less studied: to the best of our knowledge, the only papers on the topic are [26,28] by Vázquez and [19] by Gilding and Goncerzewicz (see also [6] in the p-Laplacian diffusion setting). Nonnegative solutions to (1.1) exhibit a traveling wave (TW) behaviour as t → +∞, when computed at the correct rescaled variables (see [19,Theorems 4.1 and 4.2], our main Theorem 1.4 and the change of variables (1.3)).…”

Propagation speed of degenerate diffusion equations with time delay

“…Medvedev et al [24] proved that the slowest traveling wave in the family yields the asymptotic speed of the propagation of disturbances in a class of degenerate Fisher-KPP equations. In recent works [5,6,7], more general cases of doubly nonlinear diffusion are considered, which includes both porous medium and p-Laplacian models.…”