Initial Boundary Value Problems To Two Classes Of Nonlinear Model Equations In Mathematical Physics

AbstractThe paper studies the existence, non-existence, asymptotic property and stability of global solutions of the initial boundary value problems to two classes of nonlinear model equations in mathematical physics.The paper is divided into two parts. In the first part, it discusses the existence of global weak solutions of the initial boundary value problems to a class of multidimensional quasi linear evolution equations with strong damping, nonlinear strain, nonlinear damping and source terms, which come from Viscoelasticity Mechanics. Making use of the potential well method as well as the monotonicity method, the paper arrives at the global existence and asymptotic property of weak solutions to the problems under the assumptions that the initial energy is properly small. Under the circumstances of large initial data, taking advantage of the Galerkin method a well as the compactness method, the paper gets the global weak solutions of the problems. And using the compensating energy method and the energy method respectively, the paper studies the non-existence of global weak solutions to the problems. These results implies that there exist close relations or thresholds among the growth indexes of the nonlinear terms, assuming that their growth indexes are no more than that of polynomials respectively, the value of initial energy, and the existence and non-existence of global weak solutions of the above-mentioned problems. Furthermore, the paper studies the initial boundary value problems to a class of multidimensional nonlinear evolution equations with strong damping, nonlinear strain and nonlinear external force terms. Under the assumptions that initial data is properly small, by virtue of the H -Galerkin method, the paper obtains theglobal existence, asymptotic property and stability of the classical solutions to the problemsand gains a series of new results.In the second part, the paper studies the local existence and the blowup of solutions of the initial boundary value problems to the ad?Boussinesq type equations, which arise from the theory of shallow waves and the Plasma Physics. Based on a new mathematical idea, i.e., viewing the solutions of the problems as streams starting from their initial points, by establishing a series of isometricly isomorphic Hilbert spaces, by the topological invariance of these spaces, and by exploiting the Galerkin approximation and the continuation of solutions step by step, the paper proved that under rather mild conditions for the nonlinear terms and the initial data, the initial boundary value problems of the ad?Boussinesq type equation have local generalized solutions. And by using the Jensen inequality, the comparison principle of ordinaiy differential equations, the energy method and the Fourier transform method respectively, the paper proved that under certain conditions the above-mentioned solutions will blow up in finite time.