By Serge Alinhac
The content material of this booklet corresponds to a one-semester path taught on the collage of Paris-Sud (Orsay) within the spring 1994. it truly is available to scholars or researchers with a simple straight forward wisdom of Partial Dif ferential Equations, specifically of hyperbolic PDE (Cauchy challenge, wave operator, strength inequality, finite pace of propagation, symmetric platforms, etc.). This path isn't really a few ultimate encyclopedic reference amassing all avail capable effects. We attempted as a substitute to supply a quick man made view of what we think are the most effects acquired to date, with self-contained proofs. in truth, some 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 examine. The bibliography, limited to papers the place blowup is explicitly dis stubborn, is the single half we attempted to make as whole as attainable (despite the recent preprints circulating daily) j the references are in general now not pointed out within the textual content, yet within the Notes on the finish of every bankruptcy. easy references corresponding most sensible 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 via John [J06], Strauss [St] and Zuily [Zu].
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Extra resources for Blowup for Nonlinear Hyperbolic Equations
On the other hand, there is no reason why the nonlinearity f(u) should prevent the focusing of characteristics studied in B; thus oxu may blow up at some point when t ---+ To. Can both mechanisms take place simultaneously, or which one occurs first? A first result is given by the following proposition. 1. 1) with f(u) = u 2 and initial value Uo E CJ. Assume that Uo reaches its maximum at a point XO where Uo ::> 0, u~ t= O. Then, if u exists for t < To, u remains bounded for t < To. 1. If we define ¢(X, T) for t < To to be the abscissa of the point of ordinate T on the integral curve of L starting from (X,O) and set v(X,T) = u(¢(X,T),T), then (¢,v) is a solution of PROOF OF PROPOSITION the blowup system OT¢ = V, OTV = f(v), ¢(X,O) = X, v(X,O) = uo(X).
Semilinear Wave Equations 45 then E*u 2 is unbounded in Omaxnv for any neighborhood V of (xO, to). Moreover, if then The proof will be divided into three steps. Step 1: Local existence and uniqueness in a cone a. We prove first a uniqueness result. Uniqueness lemma. (i) Let 0 be an influence domain, U1 E LOO(O), U1 = 0 for t < 0 and U2 = E * U1. Set where max means the essential supremum in x for (x, t) E O. Then m2(t) ::; lot (t - s)m1(s)ds. 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) where f (u) is a given real function. 1. Which mechanism takes place first? 1) defined for t < To. 1) reduces to a nonlinear ODE of the type studied in A, and u may blow up at time To on one of these curves. On the other hand, there is no reason why the nonlinearity f(u) should prevent the focusing of characteristics studied in B; thus oxu may blow up at some point when t ---+ To. Can both mechanisms take place simultaneously, or which one occurs first? A first result is given by the following proposition.