## 留言板

 引用本文: 沈旭辉. 具有梯度源和非局部源的反应扩散方程解的爆破时刻下界 [J]. 应用数学和力学，2022，43（4）：469-476
SHEN Xuhui. Lower Bounds for Blow-Up Time of Reaction-Diffusion Equations With Gradient Terms and Nonlocal Terms[J]. Applied Mathematics and Mechanics, 2022, 43(4): 469-476. doi: 10.21656/1000-0887.420155
 Citation: SHEN Xuhui. Lower Bounds for Blow-Up Time of Reaction-Diffusion Equations With Gradient Terms and Nonlocal Terms[J]. Applied Mathematics and Mechanics, 2022, 43(4): 469-476.

## 具有梯度源和非局部源的反应扩散方程解的爆破时刻下界

##### doi: 10.21656/1000-0887.420155

###### 作者简介:沈旭辉(1990—)，男，讲师，博士(E-mail：xhuishen@sxufe.edu.cn)
• 中图分类号: O175.29

## Lower Bounds for Blow-Up Time of Reaction-Diffusion Equations With Gradient Terms and Nonlocal Terms

• 摘要:

对于反应扩散方程解的爆破时刻研究，不仅具有理论意义，而且与安全地控制生产，控制种群密度以及环境趋化治理等实际问题密切相关。该文考虑了一类具有梯度源和非局部源的反应扩散方程解的爆破时刻下界。首先，假设区域为高维空间中的具有光滑边界的有界凸区域；其次，通过构造合适的辅助函数，利用一阶微分不等式技术和Sobolev不等式，得出解在有限时刻发生爆破时的爆破时刻下界；最后，通过两个应用实例来解释说明文中所获得的抽象结论。

•  [1] QUITTNER P, SOUPLET P H. Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States[M]. Basel: Birkhäuser Verlag AG, 2007. [2] CAFFARRELLI L A, FRIEDMAN A. Blow-up of solutions of nonlinear heat equations[J]. Journal of Mathematical Analysis and Applications, 1988, 129(2): 409-419. [3] 李远飞, 肖胜中, 陈雪姣. 非线性边界条件下具有变系数的热量方程解的存在性及爆破现象[J]. 应用数学和力学, 2021, 42(1): 92-101. (LI Yuanfei, XIAO Shengzhong, CHEN Xuejiao. Existence and blow-up phenomena of solutions to heat equations with variable coefficients under nonlinear boundary conditions[J]. Applied Mathematics and Mechanics, 2021, 42(1): 92-101.(in Chinese) [4] DING J T. Blow-up analysis of solutions for weakly coupled degenerate parabolic systems with nonlinear boundary conditions[J]. Nonlinear Analysis: Real World Applications, 2021, 61: 103315. [5] LEVINE H A. Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded Fourier coefficients[J]. Mathematische Annalen, 1975, 214: 205-220. [6] PAYNE L E, SCHAEFER P W. Lower bounds for blow-up time in parabolic problems under Neumann conditions[J]. Applicable Analysis, 2006, 85: 1301-1311. [7] 许然, 田娅, 秦瑶. 一类反应扩散方程的爆破时间下界估计[J]. 应用数学和力学, 2021, 42(1): 113-122. (XU Ran, TIAN Ya, QIN Yao. Lower bounds of the blow-up time for a class of reaction diffusion equations[J]. Applied Mathematics and Mechanics, 2021, 42(1): 113-122.(in Chinese) [8] ZHANG J Z, LI F S. Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space[J]. Zeitschrift für Angewandte Mathematik und Physik, 2019, 70: 1-150. [9] MA L W, FANG Z B. Bounds for blow-up time of a reaction-diffusion equation with weighted gradient nonlinearity[J]. Computers and Mathematics With Applications, 2018, 76(3): 508-519. [10] DAI P, MU C L, XU G Y. Blow-up phenomena for a pseudo-parabolic equation with p-Laplacian and logarithmic nonlinearity terms[J]. Journal of Mathematical Analysis and Applications, 2020, 481(1): 123-439. [11] DING J T, SHEN X H. Blow-up problems for quasilinear reaction-diffusion equations with weighted nonlocal source[J]. Electronic Journal of Qualitative Theory of Differential Equations, 2018, 75(4): 1288-1301. [12] MARRAS M, VERNIER PIRO S. Blow-up time estimates in nonlocal reaction diffusion systems under various boundary conditions[J]. Boundary Value Problems, 2017, 2017: 2. [13] SONG J C. Lower bounds for the blow-up time in a non-local reaction-diffusion problem[J]. Applied Mathematics Letters, 2011, 24(5): 793-796. [14] MARRAS M, VERNIER PIRO S, VIGLIALORO G. Lower bounds for blow-up time in a parabolic problem with a gradient term under various boundary conditions[J]. Kodai Mathematical Journal, 2014, 37(3): 532-543. [15] LACEY A A. Thermal runaway in a non-local problem modelling Ohmic beating, part 1: model derivation and some special cases[J]. European Journal of Applied Mathematics, 1995, 6(2): 127-144. [16] WANG M X, WANG Y M. Properties of positive solutions for non-local reaction-diffusion problems[J]. Mathematical Methods in the Applied Sciences, 1996, 19(14): 1141-1156. [17] MARRAS M, PINTUS N, VIGLIALORO G. On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions[J]. Discrete and Continuous Dynamical Systems (Series S), 2020, 13(7): 2033-2045. [18] EVANS L C. Partial Differential Equations[M]. Providence: American Mathematical Society, 1998. [19] LI F S, LI J L. Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary[J]. Journal of Mathematical Analysis and Applications, 2012, 385(2): 1005-1014. [20] MIZUGUCHI M, TANAKA K, SEKINE K, et al. Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains[J]. Journal of Inequalities and Applications, 2017, 2017: 299.

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##### 出版历程
• 收稿日期:  2021-06-07
• 修回日期:  2021-08-19
• 网络出版日期:  2022-03-14
• 刊出日期:  2022-04-01

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