On wave equations with supercritical nonlinearities

Abstract.

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