Blowup for Nonlinear Hyperbolic Equations by Serge Alinhac

By Serge Alinhac

The content material of this publication corresponds to a one-semester direction taught on the college of Paris-Sud (Orsay) within the spring 1994. it truly is available to scholars or researchers with a simple trouble-free wisdom of Partial Dif ferential Equations, in particular of hyperbolic PDE (Cauchy challenge, wave operator, strength inequality, finite pace of propagation, symmetric structures, etc.). This direction isn't a few ultimate encyclopedic reference accumulating all avail capable effects. We attempted in its place to supply a quick man made view of what we think are the most effects received thus far, with self-contained proofs. in reality, a few of the most crucial questions within the box are nonetheless thoroughly open, and we are hoping that this monograph will provide younger mathe maticians the will to accomplish additional study. The bibliography, constrained to papers the place blowup is explicitly dis stubborn, is the single half we attempted to make as entire as attainable (despite the hot preprints circulating daily) j the references are normally now not pointed out within the textual content, yet within the Notes on the finish of every bankruptcy. simple references corresponding top to the content material of those Notes are the books by means of Courant and Friedrichs [CFr], Hormander [HoI] and [Ho2], Majda [Ma] and Smoller [Sm], and the survey papers through John [J06], Strauss [St] and Zuily [Zu].

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7) 1 v(t) ::; C2 i° t lot (t - s)v(s)ds + h(t). sin h(C2(t - s))h(s)ds + h(t). 1 (iii) If Ui E W(C(xO, to)), i = 1,2, are two solutions of an equation U = E*u 2 +v PROOF OF THE UNIQUENESS LEMMA. 4). Blowup for Nonlinear Hyperbolic Equations 46 (ii) Let H(s) = 1 for s 2:: 0 and H(s) = 0 for s < O. Set vo(s) = v(s)H(s), hoes) = h(s)H(s), Eo(s) = sHes) and w = Eo * Vo. Then w" = Vo ~ Cw + ho. 7). (iii) We have hence because of the Cauchy-Schwarz's inequality which gives Since the E*u~ are bounded, the uniqueness lemma (i) can be applied to the function IUl - u21 2 , and (ii) of the same lemma implies then Ul = U2· ~ b.

We have thus obtained an extension of u to a larger set than Omax, a contradiction. Step 3: Additional smoothness of the solution Choose cones C and C' such that G c C' c G' c Omax. Assume Uo continuous in Omax n {t ~ a}. Then luol :::; M, in C', hence lui:::; 2M in C'. a. For to > 0 small enough, we can use the proof of the local existence and uniqueness lemma in each cone G(XO, to) c C' to see that u coincides almost everywhere in G(xO, to) with the everywhere limit u of the continuous uk. Since, as a consequence of the uniqueness lemma A.

Hence u1 (xO, to) = u 2(XO, to). We have obtained a smooth solution u in 0 1 U02' o By the definition of Omax, the supremum of all T such that the strip {t < T} is contained in Omax is just the lifespan T. Since we restricted Durselves to initial data with compact supports say, in {Ixl R}, Omax contains in fact the open set {t < T}u{lxl > t+ R}. Hence the hyperplane {t = T} intersects the complement of Omax along a compact K; using the blowup criterion, we see that (u, V'u) cannot remain in a compact subset Df G in some neighborhood of K.

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