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Self similar solutions and semi linear wave equations in

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14 pages
Self-similar solutions and semi-linear wave equations in Besov spaces Fabrice Planchon Abstract We prove that the initial value problem for semi-linear wave equations is well- posed in the Besov space _ B s p ;1 2 (R n ), where the nonlinearity is of type u p , with p 2 N and s p = n 2 2 p > 1 2 . This allows to obtain self-similar solutions as well as to recover previous results under weaker smallness assumptions on the data. Introduction We are interested in the Cauchy problem for the following semi-linear wave equation 8 < : u = u p ; u(x; 0) = u 0 (x); @ t u(x; 0) = u 1 (x); (1) for n 2. Given our main interest, scaling will play an important role: it reads 8 > < > : u 0 (x) ! u 0; (x) = 2 p1 u 0 (x) u 1 (x) ! u 1; (x) = 2 p1 +1 u 1 (x) u(x; t) ! u

  • besov spaces

  • wave equation

  • similar solution

  • linear

  • sobolev space

  • similar solutions allow

  • linear part

  • following

  • following semi-linear

  • results


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