Well-posedness Of Solutions To Two Types Of Hyperbolic Equations With Dissipative Mechanisms

The nonlinear hyperbolic equations with dissipative mechanisms are very im-portant evolution equations. They play important roles in Mathematics, Physics,and many other fields. In this dissertation, we consider two types of nonlinear hy-perbolic equations with diferent dissipative mechanisms. One is the damped waveequation with nonlinear convection. The other is conservation law with degeneratedifusion term. We investigate the global existence and large time behavior of theclassical solutions to the two types of equations. This dissertation is arranged asfollows:The first chapter introduces the dissertation. We first introduce backgroundinformation pertinent to the topic, and then review the Physics background andhistory of scientific studies on damped wave equations and degenerate hyperbolic-parabolic equations. Last, we give the specific problems we investigated in thisdissertation and summarize the main results we obtain.In Chapter2, we study the global existence and Lp(2≤p≤∞) decay esti-mates of the classical solution to the Cauchy problem of the damped wave equationwith nonlinear convection term in the multi-dimensional space. We consider a smallperturbation around constant state u. Firstly, by using the Shizuta-Kawashimacondition which describes the dissipative mechanism of nonlinear hyperbolic equa-tions, we find that the dissipative structure of the equation depends on the constantstate u and the nonlinear term of the equation. Then, we show that when somedissipative condition holds, the classical solution to the Cauchy problem existsglobally. We employ the methods introduced by Ta-tsien Li ([31]). By using thedecay properties of the solution we get the global existence directly without provinglocal existence. Since the energy estimate method can only derive the bounded es-timates of the solution, we employ the frequency decomposition methods to obtain the decay properties of the solution. We decompose the solution into two parts:a low frequency part and a high frequency part. To deal with the low frequencypart, we use Green’s function. We use the estimates in the low frequency partof Green’s function in the linearized equation and Duhamel Principle to derivethe decay properties of the low frequency part of the solution. As for the highfrequency part, we use energy estimates and Poincare′-like inequality and get thedecay rate of solution. Lastly, the Lpestimates are obtained by using the a priorestimates, decomposition method, and interpolation lemma. Since we assume theinitial datum satisfies anti-derivatives conditions, the solution decays faster thanheat kernel.In Chapter3, we continue to study the time-asymptotic behavior of the so-lution for the Cauchy problem of the damped wave equation with a nonlinearconvection term in multi-dimensional space. When the equation satisfies the dissi-pative condition, we obtained the pointwise decay estimates of the solution usingthe classical Green’s function methods. We accomplish this by firstly decomposingGreen’s function of linearized equations into three parts: low frequency, middlefrequency, and high frequency part and then derive pointwise estimates of Green’sfunction. Next, with the help of Duhamel Principle, we transform the nonlineardiferential equation into a nonlinear integral equation and get the expression ofthe solution. Then, we obtain the pointwise estimates of solutions by the a priorestimates. Given the derivative in the nonlinear term, we need to use energy es-timates to close the estimates on high derivative terms. Pointwise estimates canhelp us better understand the large time behavior of the solution. We find that thewave moves along a particular straight line and decays slowest along that straightline.In Chapter4, we investigate the classical solution to the nonlinear degeneratehyperbolic-parabolic equations. We consider the initial-boundary value problem ofthe conservation law with a degenerate difusion term, and show that the classicalsolution exists globally and decays exponentially when t goes to infinity. To obtainthe global existence of the classical solution, we employ the standard continuityargument. Firstly, we prove the local solution exists and then extend it to a global solution by uniform estimates of the solution. The main difculty is the degenera-tion of the difusion term. To deal with it, we use integration by parts, the boundeddomain, and Poincare′inequality. Meanwhile, to clarify the viscous efect of the de-generate difusion term, we introduce a modified equation whose nonlinear term’slow frequency part was cut of by a nonlocal operator. We consider the Cauchyproblem of the equation with nonlocal operator and obtain global existence and Lpdecay estimates of classical solution.