论文标题
具有混合非线性的广义三环形方程的爆炸和寿命估计值
Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities
论文作者
论文摘要
我们在本文中研究了在存在两种混合非线性的情况下对通用三角形方程的解决方案的爆炸，也就是说，我们考虑$$（tr）\ hspace {1cm} u_ {tt} -t} -t^{2m} {2m}ΔU= | U_T = | U_T | U_T | U_T | \ Mathbb {r}^n \ times [0，\ infty），带有少量的初始数据，其中$ m \ ge0 $。 $ | u |^q $）。我们在目前的工作中表明，两种非线性之间的竞争仍然为Tricomi方程$（TR）$带有$ M \ ge0 $的新打击区域，并且根据Tricomi参数$ M $，我们得出了寿命的估计。作为用于研究方程$（tr）$开发的方法，我们以不同的方法获得了与\ cite {lai2020}相同的爆炸结果，当我们仅考虑一个时间衍生的非线性时，即我们只保留$ | u_t | u_t |^p $在$（tr）$的右侧$（tr）$。
We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) \hspace{1cm} u_{tt}-t^{2m}Δu=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty),$$ with small initial data, where $m\ge0$.\\ For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $(Tr)$ with $m\ge0$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $m$. As an application of the method developed for the study of the equation $(Tr)$ we obtain with a different approach the same blow-up result as in \cite{Lai2020} when we consider only one time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand side of $(Tr)$.
