By Berti M., Biasco L.
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Extra info for Forced vibrations of wave equations with non-monotone nonlinearities
243 (2003), no. 2, 315–328. : Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Anal. 56, (2004), 1011–1046. : Cantor families of periodic solution for completely resonant wave equations, preprint SISSA 2004. [B99] J. , Univ. Chicago Press, 1999. ; Nirenberg, L. : Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math. 31 (1978), no. 1, 1–30. ; Nirenberg, L. Free vibrations for a nonlinear wave equation and a Theorem of P. Rabinowitz, Comm. Pure Appl. Math.
Berti, L. Biasco Moreover, again by Fubini Theorem, x x − ϕt (t, x) Ω h(t + x − ξ, ξ)dξ + ϕx (t, x) κ π x h(t + x − ξ, ξ)dξ dt dx κ 2π − ϕt (t, x)h(t + x − ξ, ξ) + ϕx (t, x)h(t + x − ξ, ξ) dt dξ dx = 0 κ 0 x π 2π − ϕt (t − x + ξ, x) + ϕx (t − x + ξ, x) h(t, ξ) dt dξ dx = 0 κ 0 π 2π π h(t, ξ) = 0 0 2π ξ d ϕ(t − x + ξ, x) dx dt dξ dx κ − π h(t, ξ) 0 0 0 d ϕ(t − x + ξ, x) dx dt dξ = − dx hϕ Ω and, analogously, x x Ω h(t − x + ξ, ξ)dξ dt dx = − h(t − x + ξ, ξ)dξ + ϕx (t, x) ϕt (t, x) hϕ . 79). Step 4: H(t, x) > 0 in Ω.
By the change of variables (t, x) → (t, π − x) and periodicity, 2π 2k+1 π−απ ϕˆj (t + x) − ϕˆj (t − x) dtdx ϕ1 · . . · ϕ2k+1 = απ Ωα 0 j=1 2π 2k+1 π−απ ϕˆj (t + π − x) − ϕˆj (t − π + x) dtdx = απ 0 j=1 2π 2k+1 π−απ ϕˆj (t − x) − ϕˆj (t + x) dtdx = απ 0 j=1 = (−1)2k+1 ϕ1 · . . 17). 19). 5. 20) follows by the convexity of t → t2k . 21). If b = 0 it is trivially true. If b = 0 let us divide for b2k and set x := a/b ∈ R. 21) is equivalent to prove f (x) := (x − 1)2k − x2k − 1 + 2kx2k−1 + 2kx ≥ 0.