Study Of Posed Problems On Nonlinear Heat Conduction Equations

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.