Large global solutions for energy supercritical nonlinear wave equations on \R^{3+1}

Abstract

For the radial energy-supercritical nonlinear wave equation $$\square u=-u_{tt}+△u=\pm u^7$$ on ${\mathbb{R}}^{3+1}$, we prove the existence of a class of global in forward time $C^{\infty }$-smooth solutions with infinite critical Sobolev norm $\dot{H}^{\frac{7}{6}}(\R^3)\times \dot{H}^{\f16}(\R^3)$. Moreover, these solutions are stable under suitably small perturbations . We also show that for the {\em defocussing} energy supercritical wave equation, we can construct such solutions which moreover satisfy the size condition $\Vert u(0,\cdot ){\Vert }_{L_x^{\infty }(|x|\ge 1)}>M$ for arbitrarily prescribed $M>0$. These solutions are stable under suitably small perturbations. Our method proceeds by regularization of self-similar solutions which are smooth away from the light-cone but singular on the light-cone. The argument crucially depends on the supercritical nature of the equation. Our approach should be seen as part of the program initiated in \cite{KrSchTat1}, \cite{KrSchTat2}, \cite{DoKri}.

arXiv comment38 pages

References in corpus (1)

• Self-similar solutions of semilinear wave equations with a focusing nonlinearity

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