| Abstract: | We prove that the solution operators et (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ((H)\dot]1 ?Lr+1) ×L2(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to (H)\dot]sq￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings (3], 6]). Here et(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of { |
|
?2tu-Dxu+ m2u+|u|r-1u=0, t > 0, x ? \Bbb Rn, |
| | |
| \left\{\matrix {\partial ^2_tu-\Delta _xu+ m^2u+|u|^{\rho -1}u=0, \, t>0, \, x \in {\Bbb R}^n,\cr u\vert _{t=0}(x)=\phi (x),\hfill\cr \partial _tu\vert _{t=0}(x)=\psi (x). \hfill}\right. where n 3 4, m 3 0n \geq 4, m\geq 0 and r > r* = (n+2)/(n-2)\rho >\rho ^\ast =(n+2)/(n-2) in the supercritical case. |
