【作者】 孙明岩；

【导师】 刘亚成；

【作者基本信息】 哈尔滨工程大学 ， 应用数学， 2008， 硕士

【摘要】 本文研究以下半线性双温度热传导方程u_t—△u—△u_t=u=f（u）,x∈Rⁿ,＞0,u（x,0）=u₀（x）,x∈Rⁿ.的柯西问题（初值问题）.半线性双温度热传导方程是在物理学中提出的一类非线性拟抛物方程.本文采取的研究方法主要是位势井及位势井族理论.首先,应用位势井族理论研究了解的不变集合,得到了解的真空隔离现象.其次,研究了该问题的整体弱解的存在性与解的blow-up,得到了解的整体存在性与不存在性的门槛结果.再次,运用Galerkin方法并且结合了位势井族理论研究了在临界条件J（u₀）=d下,问题的整体弱解的存在性.最后,讨论了问题解的渐近性.更多还原

【Abstract】 In this paper,we study the Cauchy problem for a class of semilinear double temperature heat equation u_t-△u-△u_t+u=f（u）,x∈Rⁿ,t＞0, u（x,0）=u₀（x）,x∈Rⁿ. Semilinear double temperature heat equation is one of nonlinear pseudoparabolic equations arised from physics.This paper adopts the method of potential well and the family of potential wells.Firstly,we study the invariant sets of solutions by using the family of potential wells and obtain vacuum isolating behavior of solutions.Secondly,we study the the existence of the global weak solutions and blow-up of solutions.We obtain a threshold result of global existence and nonexistence of solutions.Thirdly,we study the existence of the global weak solutions of the problem with critical initial condition J（u₀）=d by using the method of Galerkin and the family of potential wells.Finally,we study the asymptotic behavior of solutions of the problem.更多还原