Blow-Up of Solutions for the Coupled System of Tricomi Equations with Derivative Nonlinearity
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摘要: 在空间维数n≥2时，研究带导数非线性项的耦合Tricomi方程组的小初值问题。通过定义问题的能量解并构造适当的检验函数，得到关于解的积分泛函的不等式。根据非线性项指数的范围将解的性态研究分为次临界情形及临界情形。在次临界情形利用改进的Kato引理，在临界情形利用迭代方法，证明了问题的解会在有限时间破裂。同时，在次临界情形得到幂次形式解的生命跨度的上界估计，在临界情形得到指数形式解的生命跨度的上界估计，推广了现有文献的结论。Abstract: The small initial values problem of coupled Tricomi equations with derivative nonlinearity with space dimensional n ≥ 2 is studied.By defining the energy solutions of the problem and constructing the adequate test function,the integral functional inequalities of solutions are obtained.According to the range of nonlinearities exponents,the research process is divided into the sub-critical case and critical cases.By using the improved Kato’s lemma in the sub-critical case and iterative method in the critical case,it shows that solutions to the problem blow up in finite time.Meanwhile,the upper bound lifespan estimates in power form for the sub-critical case and exponential form for the critical case are obtained,which generalizes the conclusions of existing literatures.
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Keywords:
- derivative nonlinearity /
- coupled Tricomi equations /
- Kato’s lemma /
- iteration method /
- blowup /
- lifespan
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