導數(shù)型非線性記憶項下廣義Tricomi方程解的爆破

歐陽柏平

摘 要：考慮了一類具有導數(shù)型非線性記憶項的廣義Tricomi方程解的爆破問題。通過引入含時泛函和修正貝塞爾方程推出了解的迭代框架和第一下界；運用迭代方法證明了在次臨界情況下解的全局非存在性和生命跨度上界估計。同時，得到了導數(shù)型非線性記憶項對廣義Tricomi方程解的非局部影響。

關鍵詞：導數(shù)型非線性記憶項；廣義Tricomi方程；爆破；生命跨度

中圖分類號：O175.2

文獻標志碼：A

1 主要結果

2 主要結論的證明

參考文獻：

[1]OUYANG B P， XIAO S Z. Blow-up result for a semi-linear wave equation with a nonlinear memory term of derivative type［J］. Chinese Quarterly Journal of Mathematics， 2021， 36（3）： 235-243.

［2］ TAKAMURA H. Improved Katos lemma on ordinary differential inequality and its application to semilinear wave equations［J］. Nonlinear Anal， 2015， 125： 227-240.

［3］ TAKAMURA H， WAKASA K. The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions［J］. J Differential Equations， 2011， 251（4-5）： 1157-1171.

［4］ LAI N A， TAKAMURA H. Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity［J］. Nonlinear Analysis： Real World Applications， 2019， 45： 83-96.

［5］ ZHOU Y. Blow up of solutions to the Cauchy problem for nonlinear wave equations［J］. Chinese Ann Math Ser B， 2001， 22（3）： 275-280.

［6］ JOHN F. Blow-up for quasilinear wave equations in three dimensions［J］. Comm Pure Appl Math， 1981， 34： 29-51.

［7］ LUCENTE S， PALMIERI A. A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type[J]. Milan J Math， 2021， 89： 45-57.

［8］ CHEN W H， LUCENTE S ， PALMIERI A . Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity［J］. Nonlinear Analysis Real World Applications， 2021， 61（2）：103354.

［9］ CHEN W， PALMIERI A. Nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case［J］. Discrete Contin Dyn Syst， 2020， 40： 5513-5540.

［10］CHEN W H. Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms［J］. Nonlinear Anal， 2021， 202： 112160.

［11］CHEN W， REISSIG M. Blow-up of solutions to Nakao's problem via an iteration argument［J］. J Differential Equations， 2021， 275： 733-756.

［12］CHEN W H， PALMIERI A. Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part［J］. Z Angew Math Phys， 2019， 70（2）： 67.

［13］LAI N A， TAKAMURA H. Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey's conjecture［J］. Differ Integral Equ， 2019， 32： 37-48.

［14］HE D Y， WITT I， YIN H C. On the global solution problem for semilinear generalized Tricomi equations， I［J］. Calc Var Partial Differential Equations， 2017， 56（2）： 21.

［15］SHI J C， XIA J Y， ZHI W J. Blow-up of energy solutions for the semilinear generalized Tricomi equation with nonlinear memory term[J]. AIMS Mathematics， 2021， 6（10）： 10907-10919.

［16］RUAN Z P， WITT I， YIN H C. On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains[J]. J Differ Equ， 2015， 259： 7406-7462.

［17］HE D Y， WITT I， YIN H C. On the Strauss index of semilinear Tricomi equation［J］. Commun Pure Appl Anal， 2020， 19（10）： 4817-4838.

［18］YORDANOV B T， ZHANG Q S. Finite time blow up for critical wave equations in high dimensions［J］. Journal of Functional Analysis， 2006， 231（2）： 361-374.

［19］OLVER F W J， LOZIER D W， BOISVERT R F， et al. NIST Handbook of Mathematical Functions［M］. New York ： Cambridge University Press， 2010.

（責任編輯：于慧梅）

Blow-Up of Solutions to a Generalized Tricomi Equation with?a Nonlinear Memory Term of Derivative Type

OUYANG Baiping*

（Guangzhou Huashang College， Guangzhou 511300， China）

Abstract：In this research blow-up of solutions to a generalized Tricomi equation with a nonlinear memory term of derivative type is considered. By introducing time-dependent functionals and modified Bessel equations， an iterative frame and the first lower bound of solutions are derived. Then， application of iteration methods leads to proof of the nonexistence of global solutions and an upper bound estimate of solutions for the lifespan in subcritical case. Meanwhile， the nonlocal influence from the nonlinear memory term of derivative type on the solution of generalized Tricomi equation is obtained.

Key words：nonlinear memory term of derivative type; generalized Tricomi equation; blow-up; lifespan