一类半线性时间分数阶扩散-波动方程解的整体存在唯一性

Abstract

Abstract:

The purpose of this paper is to study the Cauchy problem of a class of semilinear time fractional diffusion-wave equations. Based on the L^r-L^q estimates obtained from the corresponding linear problem, and combined with the global iteration method, the influence of the exponential of the nonlinear term on the global existence of the solutions is studied with small data, the existence and uniqueness of global solutions are proved under certain conditions of exponential.

Key words: Time fractional diffusion-wave equation, Cauchy problem, Small data, Global solution

CLC Number:

O175.29

Xinhai He,Mei Liu,Han Yang. Existence and Uniqueness of Global Solutions for a Class of Semilinear Time Fractional Diffusion-Wave Equations[J].Acta mathematica scientia,Series A, 2022, 42(6): 1705-1718.

Trendmd

References 25

1	Fujita H . On the blowing up of solutions of the Cauchy problem for u_t=Δu + u^1+α. J Fac Sci Univ Tokyo Sect I, 1966, 13, 109- 124
2	Hayakawa K . On nonexistence of global solutions of some semi-linear parabolic differential equations. Proceedings of the Japan Academy, 1973, 49 (7): 503- 505
3	Kobayashi K , Sirao T , Tanaka H . On the growing up problem for semi-linear heat equations. Journal of the Mathematical Society of Japan, 1977, 29 (3): 407- 424
4	Amour L , Ben-Artzi M . Global existence and decay for viscous Hamilton-Jacobi equations. Nonlinear Analysis Theory Methods and Applications, 1998, 31 (5): 621- 628
5	Laurencot P , Souplet P . On the growth of mass for a viscous Hamilton-Jacobi equation. Journal d'Analyse Mathématique, 2003, 89 (1): 367- 383 doi: 10.1007/BF02893088
6	Gilding B H . The Cauchy problem for u_t=Δu +\|▽u\|^q, large-time behaviour. Journal de Mathématiques Pures et Appliquées, 2005, 84 (6): 753- 785 doi: 10.1016/j.matpur.2004.11.003
7	Kato T . Blow-up of solutions of some nonlinear hyperbolic equations. Communications on Pure and Applied Mathematics, 1980, 33 (4): 501- 505 doi: 10.1002/cpa.3160330403
8	Strauss W A. Everywhere Defined Wave Operators//Crandall M G. Nonlinear Evolution Equations. New York: Academic Press, 1978: 85-102
9	Glassey R T . Existence in the large for □u=F(u) in two space dimensions. Mathematische Zeitschrift, 1981, 178 (2): 233- 261 doi: 10.1007/BF01262042
10	Zhou Y . Cauchy problem for semilinear wave equations in four space dimensions with small initial data. J Differential Equations, 1995, 8 (2): 135- 144
11	Georgiev V , Lindblad H , Sogge C D . Weighted Strichartz estimates and global existence for semilinear wave equations. American Journal of Mathematics, 1997, 119 (6): 1291- 1319 doi: 10.1353/ajm.1997.0038
12	Tataru D . Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation. Transactions of the American Mathematical Society, 2001, 353 (2): 795- 807
13	Schaeffer J . The equation u_tt-Δu=\|u\|^p for the critical value of p. Proc Roy Soc Edinburgh Sect A, 1985, 101 (1/2): 31- 44
14	Zhou Y . Blow up of solutions to semilinear wave equations with critical exponent in high dimensions. Chinese Annals of Mathematics, 2007, 28 (2): 205- 212 doi: 10.1007/s11401-005-0205-x
15	John F . Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Mathematica, 1979, 28 (1/3): 235- 268
16	Glassey R T . Finite-time blow-up for solutions of nonlinear wave equations. Mathematische Zeitschrift, 1981, 177 (3): 323- 340 doi: 10.1007/BF01162066
17	Sideris T C . Nonexistence of global solutions to semilinear wave equations in high dimensions. Journal of Differential Equations, 1984, 52 (3): 378- 406 doi: 10.1016/0022-0396(84)90169-4
18	Voroshilov A A , Kilbas A A . The cauchy problem for the diffusion-wave equation with the caputo partial derivative. Differential Equations, 2006, 42 (5): 638- 649 doi: 10.1134/S0012266106050041
19	Zhang Q , Sun H . The blow-up and global existence of solutions of cauchy problems for a time fractional diffusion equation. Topological Methods in Nonlinear Analysis, 2015, 46 (1): 69- 92 doi: 10.12775/TMNA.2015.038
20	D'Abbicco M , Ebert M R , Picon T H . The critical exponent(s) for the semilinear fractional diffusive equation. Journal of Fourier Analysis and Applications, 2019, 25 (3): 696- 731
21	Duong P T , Mezadek M K , Reissig M . Global existence for semi-linear structurally damped σ-evolution models. Journal of Mathematical Analysis and Applications, 2015, 431 (1): 569- 596
22	Runst T , Sickel W . Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. New York: Walter de Gruyter, 1996
23	Kilbas A A , Srivastava H M , Trujillo J J . Theory and Applications of Fractional Differential Equations. New York: Elsevier Science Inc, 2006
24	肖常旺, 郭飞. 一类半线性波动方程的适定性. 数学物理学报, 2020, 40A (6): 1568- 1589 doi: 10.3969/j.issn.1003-3998.2020.06.012
	Xiao C W , Guo F . Global existence and blowup phenomena for a semilinear wave equation with time-dependent damping and mass in exponentially weighted spaces. Acta Math Sci, 2020, 40A (6): 1568- 1589 doi: 10.3969/j.issn.1003-3998.2020.06.012
25	李大潜, 周忆. 非线性波动方程. 上海: 上海科学技术出版社, 2016
	Li D Q , Zhou Y . Nonlinear Wave Equations. Shanghai: Shanghai Scientific and Technical Publisher, 2016

[1]	Su Xiao, Wang Shubin. The Cauchy Problem for the Forth Order p-Generalized Benney-Luke Equation [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1744-1753.
[2]	Chunlei He,Zihui Liu. Symmetries and Global Solutions for a Class of Hyperbolic Mean Curvature Flow [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1089-1102.
[3]	Ning-An LAI, Wei XIANG, Yi ZHOU. GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW [J]. Acta mathematica scientia,Series A, 2022, 42(3): 887-902.
[4]	Hongwu Zhang. One Regularization Method for a Cauchy Problem of Semilinear Elliptic Equation [J]. Acta mathematica scientia,Series A, 2022, 42(1): 45-57.
[5]	Xiangying Chen,Guowang Chen. Cauchy Problem for a Generalized System of Coupled Boussinesq Type Equations Arising from Nonlinear Layered Lattice Model [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1552-1567.
[6]	Changwang Xiao,Fei Guo. Global Existence and Blowup Phenomena for a Semilinear Wave Equation with Time-Dependent Damping and Mass in Exponentially Weighted Spaces [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1568-1589.
[7]	Jun Wang,Tianlu Wang,Yanhua Wen,Xianfeng Zhou. Global Solutions of IVP for N-Dimensional Nonlinear Fractional Differential Equations with Delay [J]. Acta mathematica scientia,Series A, 2018, 38(5): 903-910.
[8]	Wang Li. The Inverse Problem for the Supersonic Plane Flow Past a Curved Wedge [J]. Acta mathematica scientia,Series A, 2018, 38(4): 679-686.
[9]	Ye Yaojun, Hu Yue. Global Existence and Exponential Decay of Solution for Some Higher-Order Wave Equation [J]. Acta mathematica scientia,Series A, 2018, 38(1): 110-121.
[10]	Xie Ou, Meng Zehong, Zhao Zhenyu, You Lei. A Truncation Method Based on Hermite Functions Expansion for a Cauchy Problem of the Laplace Equation [J]. Acta mathematica scientia,Series A, 2017, 37(3): 457-468.
[11]	Deng Jiqin, Deng Ziming. Existence and Uniqueness of Solutions for Nonlocal Cauchy Problem for Fractional Evolution Equations [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1157-1164.
[12]	Zhang Hongwei, Liu Gongwei, Hu Qingying. Initial Boundary Value Problem for a Class Wave Equation with Logarithmic Source Term [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1124-1136.
[13]	Lei Qian, Li Ning, Yang Han. Blow Up and Asymptotic Behavior in the Beam Equation [J]. Acta mathematica scientia,Series A, 2015, 35(4): 738-747.
[14]	Ling Zhengqiu, Qin Siqian. Coupled Semilinear Parabolic System with Nonlinear Nonlocal Boundary Condition [J]. Acta mathematica scientia,Series A, 2015, 35(2): 234-244.
[15]	CHEN Xiang-Ying. Global Existence of Solution of Cauchy Problem for A Generalized IMBq Equation with Weak Damping [J]. Acta mathematica scientia,Series A, 2015, 35(1): 68-82.

Existence and Uniqueness of Global Solutions for a Class of Semilinear Time Fractional Diffusion-Wave Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 25

Related Articles 15

Metrics

Comments

Recommended 10