一类p-Laplace抛物方程解的全局存在性和爆破性研究  

Global Existence and Blowup of Solutions for a Class of p-Laplace Parabolic Equation

作　　者：朱立平[1] 王妍 岳红云 ZHU Liping;WANG Yan;YUE Hongyun(College of Science,Xi’an University of Architecture and Technology,Xi’an 710055)

机构地区：[1]西安建筑科技大学理学院,西安710055

出　　处：《工程数学学报》2025年第2期355-369,共15页Chinese Journal of Engineering Mathematics

基　　金：陕西省自然科学基金(2020JM-409)。

摘　　要：采用势阱法研究了一类Neumann边界条件下带非线性对数项的p-Laplace抛物方程解的有限时间爆破和全局存在性。首先,利用Galerkin方法,结合紧性原理证明了全局弱解的存在性。接着,通过能量积分和ODE不等式技巧推出了全局弱解的衰减估计。其次,引入新的辅助函数,结合凹方法得到了正初始能量条件下解在有限时间内爆破,首次给出精确的爆破时间上界估计,进而把该结果推广到了非正初始能量情况。为了更直观地说明该理论结果,最后给出数值算例模拟了解的长时间衰减行为以及不同初始能量级下解的爆破性质,说明了参数p对解的演化的影响,验证了理论分析的正确性。In this paper,by using the potential well method,finite time blowup and global existence of solutions are studied for a class of p-Laplace parabolic equations with nonlinear logarithmic term and Neumann boundary conditions.Firstly,the existence of global weak solutions is proved by the Galerkin method and the compactness principle.Then,the decay estimation of the global weak solution is derived by an energy integration and ODE inequality technique.Secondly,through constructing an auxiliary function and combining the concave method,thefinite time blowup of the solution under the condition of positive initial energy is obtained,and the accurate upper bound estimation of blowup time is given for thefirst time.Furthermore,the results are extended to the case of non-positive initial energy.In order to more intuitively explain the theoretical results,some numerical examples are given to show the long-time decay behavior and the blowup properties of solutions under different initial energy levels,the effects of parameter p on the evolution of solutions and the correctness of the theoretical results.

关 键 词：p-Laplace 对数非线性 势阱法 全局存在性 爆破 

分 类 号：O175.26[理学—数学]