Felli V., Schneider S.'s A note on regularity of solutions to degenerate elliptic PDF

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

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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. [2] 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.

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A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type by Felli V., Schneider S.


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