Solutions blowing up on any given compact set for the energy subcritical wave equation

Abstract

We consider the focusing energy subcritical nonlinear wave equation ${\partial }_{tt}u-\Delta u=|u{|}^{p-1}u$ in ${\mathbb{R}}^N$, $N\ge 1$. Given any compact set $E\subset {\mathbb{R}}^N$, we construct finite energy solutions which blow up at $t=0$ exactly on $E$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x)=\kappa (t+A(x){)}^{-\frac{2}{p-1}}$, where $A\ge 0$ vanishes exactly on $E$, which is a solution of the ODE $h^{\prime \prime }=h^p$. We refine this first ansatz inductively using only ODE techniques and taking advantage of the fact that (for suitably chosen $A$), space derivatives are negligible with respect to time derivatives. We complete the proof by an energy argument and a compactness method.

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Cited by in corpus (1)

• Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient