By Felli V., Schneider S.
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Extra info for A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type
Existence for a nonlinear wave equation 43 We can now give the Proof of Corollary 1. 7). Assume for contradiction that the life–span Tmax of the solution is finite. Then, since 1 ≤ α(t) ≤ 1, (1 + Tmax )β all the assumptions (Q1) , (Q2) –(Q6) (recall that m ≥ 2) are fulfilled for T = Tmax . From Theorem 8 we get a contradiction, which complete the proof. We finish the section with the proof of the following compactness lemma, used above in the proof of local existence. Lemma 2. Let Ω ⊂ Rn be bounded and 2 < r0 < r, where r = 2n/(n − 2) if n ≥ 3, r = ∞ if n = 1, 2.
Next, for s, t ∈ [0, T ], ηε φ(s) − ηε φ(t) 2 ≤ φ(s) − φ(t) 2 . Consequently, applying the Arzel´ a–Ascoli Theorem again, it follows that (up to a subsequence), ηε φ(t) − φ(t) 2 → 0 uniformly in [0, T ] as ε → 0. Since φε (t) − φ(t) 2 ≤ ηε (ρε ∗ φ)(t) − ηε φ(t) ≤ ρε ∗ φ(t) − φ(t) for all t ∈ [0, T ], we derive φε → φ in proof. 2 2 + ηε φ(t) − φ(t) + ηε φ(t) − φ(t) C([0, T ]; L2 (Rn )). A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.  A. Ambrosetti and G. Prodi, “A Primer of Nonlinear Analysis,” Cambridge Univ.
A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolutions equations with dissipation, Arch. Rational Mech. , 137 (1997), 341–361. A. Levine and G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. , 129 (2001), 693–805. L. Lions, “Quelques m´ethodes de r´esolutions des probl`emes aux limites non lin´eaires,” Dunod, Paris, 1969. L. Lions and W. A. Strauss, On some nonlinear evolution equations, Bull.
A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type by Felli V., Schneider S.