By Michael Reissig, Bert-Wolfgang Schulze
This booklet offers a number of contemporary advancements within the thought of hyperbolic equations. The rigorously chosen invited and peer-reviewed contributions take care of questions of low regularity, severe development, ill-posedness, decay estimates for ideas of alternative non-linear hyperbolic versions, and introduce new techniques according to microlocal tools.
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Extra info for New Trends in the Theory of Hyperbolic Equations
For any α ∈ (0, 1) and T > 0 we have f ∈ L1 ((0, T ); H 0 rad (Ωα (t))). Proof. As before we use the notation rα . 14) for Q we have: z= f= 1 t2 zQ + sin 2Q 2α2 . 20) Wave Maps 45 where in terms of z we have 2 z α −1 , α 1 + z α2 1 Q (z) = so we get 1 f (t, ·) 2 L2 rad (Ωα (t)) ≤ ≤ ≤ C t4 2 tα zα 2 (1 + z α )2 0 ⎛ 1 tα C⎝ t4 ⎛ C⎝ t4 z + (sin 2Q)2 tα r dr + 2 1 tα r3 2 where we have used the inequality (sin 2Q)2 r dr⎠ 0 ⎞ tα 0 ⎞ 1 2 α (1 + z α )2 0 r dr 1 2 1+z α C dr + t α ⎠ = 2 , 2 t4− α ≤ 1 and C = C(α, T ) > 0 is a constant independent of t.
126) are invariant when exchanging u1 and u2 , there is no loss of generality in assuming that |ττ1 | + |ξ1 | ≥ |ττ2 | + |ξ2 |. 134) and h(τ, ξ) = |τ | + |ξ| s |τ | − |ξ| θ . 125) will be established if we can verify the inequality |τ | + |ξ| s |τ | − |ξ| θ u1 u2 (τ, ξ) L2τ,ξ ≤ C F1 L2τ,ξ F2 L2τ,ξ . 136) and obtain I L2τ,ξ ≤ C F1 L2τ ≤ C F1 L2τ 1 ,ξ1 1 ,ξ1 |F F2 (ττ2 , ξ2 )| dξ2 dττ2 h(ττ2 , ξ2 ) 1 F2 L2τ ,ξ . 2 2 h2 L 2 τ1 ,ξ1 The conditions s > n/2, θ > 1/2 imply that 1 h2 < ∞, L2τ 1 ,ξ1 so that I L2τ,ξ ≤ C F1 F2 L2 L2 .
Proof. 27) of the preceding lemma. 3. Estimates for the homogeneous problem The spaces H s,θ can be regarded as Sobolev spaces “adapted” to the wave equation, as the following properties show. Consider the linear homogeneous wave equation utt − ∆u = 0, t ∈ R, x ∈ Rn u(0, x) = u0 , ut (0, x) = u1 (x). 31) where 0 ≤ t ≤ T0 and C > 0, imply for any s ∈ R and T0 > 0 the existence of a solution u0 ∈ H s , u1 ∈ H s−1 ⇒ u ∈ C([0, T0 ]; H s ). 19), and T > 0. 33). 4. Assume s ≥ 0, θ > 0. 35) H s−1,θ for all 1 N >s+θ+ .