论文标题
与Glassey猜想有关的广义Tricomi方程的爆炸和寿命估计值
Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture
论文作者
论文摘要
我们在本文中研究了针对派生类型$ u_ {tt} -t^{2M}ΔU= | u_t |^p $ for $ m \ ge0 $的半连通三角米方程的小数据cauchy问题。从上方制定的爆炸结果和寿命估计值$ 1 <p \ le 1+ \ frac {2} {（m+1）（n-1）（n-1）-m} $。如果$ m = 0 $，我们的结果与半连续波方程的结果一致。新颖性在于结合截止函数，修改的贝塞尔函数和谐波函数来构建新的测试功能。有趣的是，如果$ n = 2 $，则爆破功率独立于$ m $。我们还提供了局部存在的结果，这意味着至少在$ 1 $维的情况下，寿命估算的最佳性。
We study in this paper the small data Cauchy problem for the semilinear generalized Tricomi equations with a nonlinear term of derivative type $u_{tt}-t^{2m}Δu=|u_t|^p$ for $m\ge0$. Blow-up result and lifespan estimate from above are established for $1<p\le 1+\frac{2}{(m+1)(n-1)-m}$. If $m=0$, our results coincide with those of the semilinear wave equation. The novelty consists in the construction of a new test function, by combining cut-off functions, the modified Bessel function and a harmonic function. Interestingly, if $n=2$ the blow-up power is independent of $m$. We also furnish a local existence result, which implies the optimality of lifespan estimate at least in the $1$-dimensional case.
