Global existence for 2-D wave maps equation in exterior domains

Abstract

In the paper [H. Kubo, Global existence for exterior problems of semilinear wave equations with the null condition in 2D, Evol. Equ. Control Theory 2 (2013), no. 2, 319-335], for the 2-D semilinear wave equation system $({\partial }_t^2-\Delta )v^I=Q^I({\partial }_{t}v,{\nabla }_{x}v)$ ($1\le I\le M$) in the exterior domain with Dirichlet boundary condition, it is shown that the small data smooth solution $v=(v^1,\cdot \cdot \cdot ,v^M)$ exists globally when the cubic nonlinearities $Q^I({\partial }_{t}v,{\nabla }_{x}v)=O(|{\partial }_{t}v{|}^3+|{\nabla }_{x}v{|}^3)$ satisfy the null condition. We now focus on the global Dirichelt boundary value problem of 2-D wave maps equation with the form $\square u^I={\sum }_{J,K,L=1}^M{C}_{IJKL}{u}^J{Q}_0(u^K,u^L)$ $(1\le I\le M)$ and $Q_0(f,g)={\partial }_{t}f{\partial }_{t}g-{\sum }_{j=1}^2{\partial }_{j}f{\partial }_{j}g$ in exterior domain. By establishing some crucial classes of pointwise spacetime decay estimates for the small data solution $u=(u^1,\cdot \cdot \cdot ,u^M)$ and its derivatives, the global existence of $u$ is shown.

arXiv admin note: text overlap with arXiv:2411.06984

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