非线性热传导方程适定性研究
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摘要
本文研究针对热传导过程建立的几类非线性偏微分方程的适定性问题，应用固体力学理论研究了薄板在热流冲击作用下的非Fourier效应。这些问题的研究对于揭示热传导过程的物理本质，推动对热传导问题基础理论、数值方法和模拟仿真技术的研究是有益的。
本文首先研究了一类反映热传导的半线性抛物方程的柯西问题，通过引进位势井族，得到了位势井族的相关性质和解的不变集合。结合Galerkin近似解方法及能量估计法，得到了相应问题整体弱解的存在性结论，同时证明了这个整体解随着时间趋于无穷而衰减到零。利用位势井方法及凸性方法，证明了初值在不稳定集合时相应问题解的有限时间爆破。同时，针对所研究问题的临界现象，利用位势井族方法，结合能量估计法及凸性方法，得到了整体弱解存在及有限时间爆破的结论。
本文继而讨论了一类反映热波传播的半线性波动方程的柯西问题，在研究具有单一源项的波动方程的初边值问题的基础上，进一步研究了多个源项的波动方程的初边值问题整体弱解的存在性。本文针对此问题重新定义了位势井和位势井族，通过初等的方法找到了位势井深度函数的计算方法，讨论了位势井族的相关性质和解的不变集合。特别是，针对新的位势井方法给出了计算机描述位势井内部结构的可行性，并将此手段程式化，以期解决更复杂的同类问题。本文结合Galerkin近似解方法及能量估计法，得到了相应问题整体弱解的存在性结论，在适当的条件下，利用位势井方法及凸性方法，讨论了相应问题解的有限时间爆破。同时，针对所研究问题的临界现象，利用位势井族的方法，结合能量估计法及凸性方法，得到了整体弱解存在及有限时间爆破等一系列结论。
本文对带有幂次组合型非线性项的非线性波动方程初边值问题进行了探讨。首先针对此类问题定义了新的位势井结构，而后利用第3章给出的分析技术对复杂的非线性项对位势井结构的影响进行分析，进而得出复杂的非线性项影响下的问题解的性质，证明了低能状态和临界状态下解的整体存在性和有限时间爆破。
针对有限厚度板的传热问题，在考虑了非傅立叶效应的基础上，建立并研究了在突加常热流边界条件（第二类传热边界条件)下的双曲传热模型。应用积分变换的方法，得到了瞬态温度场问题的解析解。利用数值分析的手段，讨论了非傅立叶导热效应在有限厚度板内的具体行为，通过与基于傅立叶经典导热理论所得数值结果进行了比对，阐明了它们之间的异同以及两类不同模型的适用范围。同时，考察了热冲击作用下的简支矩形板，对其挠度的变化规律进行了详细的讨论，给出了薄板内应力场的分布。该方法简洁有效，可以推广应用到热冲击作用下的结构强度分析领域中。
This work studies the well-posedness several classes of nonlinear partial differentialequations arising in the heat conduction process and applies the solid mechanics theory tostudy the non-Fourier effect of a sheet under heat shock. These studies are meaningful forrevealing the physical nature of the heat conduction process and promoting the basic theoryof heat conduction problems， numerical methods and the simulation techniques.
This work first studies the Cauchy problem for a class of semi-linear parabolic equationsreflecting the heat conduction. By introducing the family of potential wells we get thecorresponding properties of the potential wells and the invariant sets. Combining with theGalerkin approximate method and the energy estimate we get the existence results of globalweak solutions of the corresponding problem. And we also prove that the global solution astime tends to infinity， decays to zero. Using potential well method and convexity method weprove that the initial value in an unstable manifold may lead that the corresponding solutionsblow up in finite time. For critical case of the initial energy， by the potential wells methodcombined with the energy estimation method and convexity method， we prove the existenceof the global weak solutions and the finite time blow up of the solutions.
Then we discuss the Cauchy problem for a class of the semi-linear wave equationsreflecting the heat wave propagation. First we try the initial boundary value problem ofweave equations with a single source term，which is based to study further the initialboundary value problem of weave equations with several source terms. For this problem were-define the potential wells and the family of potential wells. By the primary method we findthe way of calculating the potential well depth function and discuss the properties of thefamily of potential wells and the invariant sets. In particular， for the newly defined potentialwell we give computer description of the internal structure of the potential well and make thisprocess programmable in order to solve the similar problems with more complex terms. Inthis work， we apply the Galerkin approximate method and the energy estimation， we provethe existence results of global weak solutions for the corresponding problem. Underappropriate conditions， by the potential well method and the convexity method， we obtainthe finite time blow up for the corresponding solutions. For the critical case of the problems，by the family of potential wells combined with the energy estimation method and theconvexity method，we prove the existence of the global weak solutions and the finite timeblow of solutions.
We also consider the initial boundary value problem of the nonlinear wave equations with the combined power type nonlinear terms. For such problem we define a new potentialwell and then use the analysis techniques given in the previous chapter to analyze the affectsof the complex nonlinear terms to the potential well structure. And then we come to thenature of the solution under the complex non-linear effect to prove that the solutions in thelow-energy state and the critical state exist globally or blow up in finite time.
In this work，for the finite thickness plate we take into account the non-Fourier effect ofheat transfer problems and establish hyperbolic non-Fourier heat transfer model with thesecond boundary conditions for the heat transfer boundary conditions (suddenly appliedconstant heat flux). Applying the integral transform method to calculate the transienttemperature field we obtain the analytical soluyhution. By numerical analysis， we analyzethe non-Fourier heat conduction behavior in the finite thickness plate. Then we compare itwith the results by the classical Fourier law of heat conduction in order to clarify thedistinctions between the two models， as well as give the uses of the two models. Inaddition， detailed discussion of the deflection of a simply supported rectangular plate underthermal shock and the distribution of stress field in the metal plate under thermal shockloading are given. The above analysis method is simple and effective， which can be appliedto the calculation of the strength of the structure in the thermal shock effect.
本文首先研究了一类反映热传导的半线性抛物方程的柯西问题，通过引进位势井族，得到了位势井族的相关性质和解的不变集合。结合Galerkin近似解方法及能量估计法，得到了相应问题整体弱解的存在性结论，同时证明了这个整体解随着时间趋于无穷而衰减到零。利用位势井方法及凸性方法，证明了初值在不稳定集合时相应问题解的有限时间爆破。同时，针对所研究问题的临界现象，利用位势井族方法，结合能量估计法及凸性方法，得到了整体弱解存在及有限时间爆破的结论。
本文继而讨论了一类反映热波传播的半线性波动方程的柯西问题，在研究具有单一源项的波动方程的初边值问题的基础上，进一步研究了多个源项的波动方程的初边值问题整体弱解的存在性。本文针对此问题重新定义了位势井和位势井族，通过初等的方法找到了位势井深度函数的计算方法，讨论了位势井族的相关性质和解的不变集合。特别是，针对新的位势井方法给出了计算机描述位势井内部结构的可行性，并将此手段程式化，以期解决更复杂的同类问题。本文结合Galerkin近似解方法及能量估计法，得到了相应问题整体弱解的存在性结论，在适当的条件下，利用位势井方法及凸性方法，讨论了相应问题解的有限时间爆破。同时，针对所研究问题的临界现象，利用位势井族的方法，结合能量估计法及凸性方法，得到了整体弱解存在及有限时间爆破等一系列结论。
本文对带有幂次组合型非线性项的非线性波动方程初边值问题进行了探讨。首先针对此类问题定义了新的位势井结构，而后利用第3章给出的分析技术对复杂的非线性项对位势井结构的影响进行分析，进而得出复杂的非线性项影响下的问题解的性质，证明了低能状态和临界状态下解的整体存在性和有限时间爆破。
针对有限厚度板的传热问题，在考虑了非傅立叶效应的基础上，建立并研究了在突加常热流边界条件（第二类传热边界条件)下的双曲传热模型。应用积分变换的方法，得到了瞬态温度场问题的解析解。利用数值分析的手段，讨论了非傅立叶导热效应在有限厚度板内的具体行为，通过与基于傅立叶经典导热理论所得数值结果进行了比对，阐明了它们之间的异同以及两类不同模型的适用范围。同时，考察了热冲击作用下的简支矩形板，对其挠度的变化规律进行了详细的讨论，给出了薄板内应力场的分布。该方法简洁有效，可以推广应用到热冲击作用下的结构强度分析领域中。
This work studies the well-posedness several classes of nonlinear partial differentialequations arising in the heat conduction process and applies the solid mechanics theory tostudy the non-Fourier effect of a sheet under heat shock. These studies are meaningful forrevealing the physical nature of the heat conduction process and promoting the basic theoryof heat conduction problems， numerical methods and the simulation techniques.
This work first studies the Cauchy problem for a class of semi-linear parabolic equationsreflecting the heat conduction. By introducing the family of potential wells we get thecorresponding properties of the potential wells and the invariant sets. Combining with theGalerkin approximate method and the energy estimate we get the existence results of globalweak solutions of the corresponding problem. And we also prove that the global solution astime tends to infinity， decays to zero. Using potential well method and convexity method weprove that the initial value in an unstable manifold may lead that the corresponding solutionsblow up in finite time. For critical case of the initial energy， by the potential wells methodcombined with the energy estimation method and convexity method， we prove the existenceof the global weak solutions and the finite time blow up of the solutions.
Then we discuss the Cauchy problem for a class of the semi-linear wave equationsreflecting the heat wave propagation. First we try the initial boundary value problem ofweave equations with a single source term，which is based to study further the initialboundary value problem of weave equations with several source terms. For this problem were-define the potential wells and the family of potential wells. By the primary method we findthe way of calculating the potential well depth function and discuss the properties of thefamily of potential wells and the invariant sets. In particular， for the newly defined potentialwell we give computer description of the internal structure of the potential well and make thisprocess programmable in order to solve the similar problems with more complex terms. Inthis work， we apply the Galerkin approximate method and the energy estimation， we provethe existence results of global weak solutions for the corresponding problem. Underappropriate conditions， by the potential well method and the convexity method， we obtainthe finite time blow up for the corresponding solutions. For the critical case of the problems，by the family of potential wells combined with the energy estimation method and theconvexity method，we prove the existence of the global weak solutions and the finite timeblow of solutions.
We also consider the initial boundary value problem of the nonlinear wave equations with the combined power type nonlinear terms. For such problem we define a new potentialwell and then use the analysis techniques given in the previous chapter to analyze the affectsof the complex nonlinear terms to the potential well structure. And then we come to thenature of the solution under the complex non-linear effect to prove that the solutions in thelow-energy state and the critical state exist globally or blow up in finite time.
In this work，for the finite thickness plate we take into account the non-Fourier effect ofheat transfer problems and establish hyperbolic non-Fourier heat transfer model with thesecond boundary conditions for the heat transfer boundary conditions (suddenly appliedconstant heat flux). Applying the integral transform method to calculate the transienttemperature field we obtain the analytical soluyhution. By numerical analysis， we analyzethe non-Fourier heat conduction behavior in the finite thickness plate. Then we compare itwith the results by the classical Fourier law of heat conduction in order to clarify thedistinctions between the two models， as well as give the uses of the two models. Inaddition， detailed discussion of the deflection of a simply supported rectangular plate underthermal shock and the distribution of stress field in the metal plate under thermal shockloading are given. The above analysis method is simple and effective， which can be appliedto the calculation of the strength of the structure in the thermal shock effect.
引文
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