Existence and Number of Global Solutions to Model Nonlinear Partial Differential Equations
In this dissertation we studied nonlinear partial differential equations in two different directions. We apply the bifurcation theory to investigate a number of positive solutions of the semilinear Dirichlet boundary value problem on a n-dimensional ball for the second order elliptic equation with periodic nonlinearity containing a positive parameter. Our approach appeals to the well known results of B. Gidas, W.-M. Ni, L. Nirenberg, the bifurcation theorems of M. G. Crandall and P. H. Rabinowitz, and the stationary phase method. Further, we investigate the issue of global existence of the solutions of the Cauchy problem for the semilinear Tricomi-type equations, appearing in the boundary value problems problems of gas dynamics. We study Cauchy problem trough integral equation and give some sufficient conditions for the existence of the global weak solutions. We prove necessity of these conditions. We obtain necessary condition for the existence of the self-similar solutions for the semilinear Tricomi-type equation. In our approach we employ the fundamental solution and the Lp-Lq estimates for the linear Tricomi-type equations.
School:University of Cincinnati
School Location:USA - Ohio
Source Type:Master's Thesis
Keywords:bifurcation phenomenon multiplicity of solutions solution curve oscillatory integrals tricomi type equations global existence lp lq estimates
Date of Publication:01/01/2005