On Center Subspace Behavior in Thin Film Equations

The large-time behavior of weak nonnegative solutions of the thin film equation (TFE) with absorption $u_t=-\nabla\cdot(|u|^n\nabla\Delta u)-|u|^{p-1}u$, with parameters $n\in(0,3)$ and $p>1$, is studied. The standard free-boundary problem (FBP) with zero height, zero contact angle, and zero-flux conditions at the interface and bounded compactly supported initial data is considered. It is shown that there exists the critical absorption exponent $p_0=1+n+\frac{4}{N}$ such that, for $p=p_0$, the asymptotic behavior of solutions $u(x,t)$ for $t\gg1$ is represented by the well-known source-type solution of the pure TFE absorption, $u_s(x,t)=t^{-\beta N}F(y)$, $y=x/t^{\beta}$, with the exponent $\beta=\frac{1}{4+nN}$, which is perturbed by a couple of $\ln t$-factors. For $n=1$, this behavior is associated with the center subspace for the rescaled linearized thin film operator and is given by $u(x,t)\sim\nobreak(t\ln t)^{-\beta N}F(x/t^{\beta}(\ln t)^{-\beta N /4})$, with $\beta=\frac{1}{4+N}$, where $F(y)=\frac{1}{8(N+2)(N+4)}(a_*^2-|y|^2)^2$ and the constant $a_*>0$ depends on dimension $N$ only. The $2m$th-order generalization of such TFEs with critical absorption is considered, and some local and asymptotic features of changing sign similarity solutions of the Cauchy problem are described. Our study is motivated by the phenomenon of logarithmically perturbed source-type behavior for the second-order porous medium equation (PME) with critical absorption $u_t=\nabla\cdot(u^n\nabla u)-u^p$ in $\mathbb{R}^N\times\mathbb{R}_+$, $p_0=1+n+\frac{2}{N}$, $n\geq0$, which has been known since the 1980s.