Critical exponent for the semilinear wave equations with a damping increasing in the far field
Critical exponent for the semilinear wave equations with a damping increasing in the far field
复制标题
DOI：
10.1007/s00030-018-0546-2
复制
发表时间：
2018-09
期刊：
影响因子：
--
通讯作者：
K. Nishihara;M. Sobajima;Yuta Wakasugi
中科院分区：
文献类型：
--
作者：
K. Nishihara;M. Sobajima;Yuta Wakasugi
我们考虑带有阻尼项 u tt-Δ u+ c (t, x) ut=| 的半线性波动方程的柯西问题你| p,(t, x)ε(0,∞)× RN, u (0, x)= ε u 0 (x), ut (0, x)= ε u 1 (x), xε RN，其中 p> 1 并且阻尼项的系数具有形式 c (t, x)= a 0 (1+| x| 2)-α/2 (1+ t)-β，其中 a 0> 0、α< 0， βε(-1, 1]。特别地，我们主要考虑α< 0，β= 0或α< 0，β= 1的情况，这意味着α+ β< 1，即阻尼在空间上递增且有效。我们的目的是证明临界指数由p= 1+ 2 N-α给出。这表明临界指数与相应的抛物线方程c(t, x)的临界指数相同vt-Δ v=| v| p 通过指数型权重函数的加权能量估计和 Caffarelli-Kohn-Nirenberg 不等式的特例来证明，爆炸部分则通过 Ikeda 和 Sobajima 引入的测试函数方法来证明。
We consider the Cauchy problem of the semilinear wave equation with a damping term u tt-Δ u+ c (t, x) ut=| u| p,(t, x)∈(0,∞)× RN, u (0, x)= ε u 0 (x), ut (0, x)= ε u 1 (x), x∈ RN, where p> 1 and the coefficient of the damping term has the form c (t, x)= a 0 (1+| x| 2)-α/2 (1+ t)-β with some a 0> 0, α< 0, β∈(-1, 1]. In particular, we mainly consider the cases α< 0, β= 0 or α< 0, β= 1, which imply α+ β< 1, namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by p= 1+ 2 N-α. This shows that the critical exponent is the same as that of the corresponding parabolic equation c (t, x) vt-Δ v=| v| p. The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli–Kohn–Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima. We also give an upper estimate of the lifespan.