A model problem for groundwater flow with dynamic capillary pressure: stability of travelling waves
The authors consider a vertical nonsteady groundwater flow with dynamic capillary pressure. The governing equation is of a diffusion-convection type, with a nonlinear third-order term. It is investigated a special case when the original problem reduces to a pseudo-parabolic Burgers equation \[ u_t= u_{xx}+ 2uu_x+ \varepsilon^2 u_{xxt}\quad\text{on }\mathbb{R}\times [0, T], \] with the initial condition \[ u(x)= u_0(x)\quad\text{in }\mathbb{R}, \] where \(\varepsilon> 0\) is a parameter. A well-posedness result for the initial value problem and the stability of travelling wave solutions are proved.
scientific article; zbMATH DE number 1880004