Concentration of Local Energy for Two-dimensional Wave Maps

Abstract

We construct some particular kind of solution to the two - dimensional equivariant wave map problem with inhomogeneous source term in space-time domain of type $\Omega_\alpha(t) = {x \in \mathbb R^2 : |x|^\alpha < t}$, where $\alpha\in (0, 1]$. More precisely, we take the initial data $(u_0, u_1)$ at time T in the space $H^{1+\epsilon} \times H^\epsilon$ with some $\epsilon > 0$. The source term is in $L^1((0, T); H^\epsilon(\Omega_\alpha(t)))$ and we show that the $H^{1+\epsilon}$ -norm of the solution blows-up, when $t \rightarrow 0_+$ and $\alpha\in (0, 1 − \epsilon)$.