论文标题
具有远场退化的准电波方程的局部溶解性：1D情况
Local solvability for a quasilinear wave equation with the far field degeneracy: 1D case
论文作者
论文摘要
我们研究了quasilinear Wave方程的Cauchy问题$ \ partial^2 _t u = u^{2a} \ partial^2_x u + f（u + f（u）u_x $带有$ a \ geq 0 $，并在新条件下及时显示本地及时存在的结果。在上一个结果中，假定$ u（0，x）\ geq C_0> 0 $对于某些常数$ C_0 $证明存在和唯一性。该假设确保方程不会退化。在本文中，我们允许方程在空间无穷大时退化。也就是说，我们认为$ u（0，x）> 0 $和$ u（0，x）\ rightarrow 0 $ as $ | x | \ rightarrow \ infty $。此外，为了证明局部适应性良好，我们发现出现了所谓的Levi条件。我们的证明是基于特征的方法和通过加权$ l^\ infty $估计的收缩原理的方法。
We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0>0$ for some constant $c_0$ to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \infty$. Furthermore, to prove the local well-posedness, we find that the so-called Levi condition appears. Our proof is based on the method of characteristic and the contraction mapping principle via weighted $L^\infty$ estimates.
