Papers Published
Abstract:
In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions d ≥ 1 J(h): = (∫ℝd|h|dx)a-1 ∫ℝd |∇h|2 dx/(∫ℝd |h|m+1 dx)a+1/m+1 ≥ β0, where m > 0 for d = 1, 2, 0 < m < d+2/d-2 for d ≥ 3, and a = d+2(m+1)/md. The Euler-Lagrange equation for critical points of J(h) in the non-negative radial decreasing function space is given by a free boundary problem for a generalized Lane-Emden equation, which has a unique solution (denoted by hc) and the solution determines the best constant for the above generalized Sz. Nagy inequality. The connection between the critical mass Mc = ∫Rdbl; hc dx = 2√2π/3 for the thin-film equation and the best constant of the Sz. Nagy inequality in one dimension was first noted by Witelski et al (2004 Eur. J. Appl. Math. 15 223-56). For the following critical thin film equation in multi-dimension d ≥ 2 ht + ∇ · (h ∇ Delta; h) + ∇ · (h ∇ hm) = 0, x ϵ ℝd, where m = 1 + 2/d, the critical mass is also given by Mc:= ∫ℝd hc dx. A finite time blow-up occurs for solutions with the initial mass larger than Mc. On the other hand, if the initial mass is less than Mc and a global non-negative entropy weak solution exists, then the second moment goes to infinity as t → ∞ or h(·, tk) ⇀ 0 in L1(ℝd) for some subsequence tk → ∞. This shows that a part of the mass spreads to infinity.