(untitled)
We study capillary spreadings of thin films of liquids of power-law rheology. These satisfy u_t+(u^λ+2|u_xxx^λ-1u_xxx)_x=0, where u(x,t) represents the thickness of the one-dimensional liquid and λ > 1. We look for traveling wave solutions so that u(x,t)=g(x+ct) and thus g satisfies g'''=|g-ε|^1λg^1+2λ sgn(g-ε). We show that for each ε > 0 there is an infinitely oscillating solution, g_ε, such that lim_t→∞g_ε=ε and that g_ε→g_0 as ε→ 0, where g_0 ≡ 0 for t ≥ 0 and g_0=c_λ|t|^3λ2λ+1 for t < 0 for some constant c_λ. , number=1, journal=Journal of Partial Differential Equations, year=2018, month=Aug., pages=33–67