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On uniqueness for semilinear wave equations

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17 pages
On uniqueness for semilinear wave equations Fabrice Planchon 01/28/2002 Abstract We prove that uniqueness holds in C t ( _ H s ) for solutions of u = u p for suitable values of s; p. This includes the _ H 1 critical case for n = 4; 5; 6. Introduction Let us consider the equation u = u 3 ;(1) with Cauchy data (u 0 ; u 1 ) 2 ( _ H 1 ; L 2 ) in space dimension n = 4. This equa- tion is known to be (locally) well-posed in the energy space, yielding a solu- tion u which is C t ( _ H 1 ). This solution can be extended to a global one in the defocusing case ([12]). However uniqueness holds only with some additional assumptions, like u 2 L 3 t (L 6 x ). This restriction can be related to the choice of spaces involved in the xed point procedure yielding a solution. Hence a natural question would be whether such assumptions are necessary to pro- vide uniqueness, or if uniqueness in the energy class holds.

  • energy class

  • nonlinearity like

  • nonlinearities

  • however uniqueness

  • has dimension

  • holds only

  • ask whether

  • such

  • wave equations


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