ILL POSEDNESS FOR THE 2-D WAVE MAP EQUATION

The well posedness of the Cauchy problems for the 2+1 dimensional wave equation with the two dimensional sphere as target is investigated. We prove that the solutions are not unique in Sobolev spaces below the critical exponent and also in suitable Besov spaces of critical regularity. We also investigate ill-posedness in the critical Sobolev space of order 1, showing that the problem is not well-posed in presence of a forcing term.