The Cauchy Problem For A Kind Of Nonlinear Dissipative Wave Equations With Variable Coefficients

The Cauchy problem of nonlinear dissipative wave equations with variable coefficients is considered in this paper, where ε>0, coefficients α(x)∈C0(R"),b(x)∈C1(R") are positive functions. Such a system appears in models for traveling waves in a non-homogeneous gas. Firstly, the global existence of problem (0.1) is studied, furthermore, the decay estimates for the energy, L2 and Lp+1 norms of the solutions are established. In addition, the blow-up of the solutions is also studied.In the Chapter 1, some historical background and some known results for wave equations are introduced respectively.In the Chapter 2, a new weighted energy function is defined by introducing the multipliers, then the estimate of the energy inequality is established combining with the weighted energy method.In the Chapter 3, for sufficiently small data with compact support, if the power p of nonlinear is larger than the expect exponent, we prove that there exists a global solution by using the multiplier method and weighted energy method. Furthermore, the precise decay estimates for the energy, L2 and Lp+1 norms of the solutions are also established.In the Chapter 4, by constructing a test function, the blow up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinear less than some constant.