Singular Limit Of Solutions Of The Nonlinear Parabolic Equations

Singular limit of solutions of the nonlinear parabolicequationThis thesis reviews nonlinear parabolic equations of the singular limit. We first considered the singular limit for the Canchy problem of the p-Laplacian equation when the initial value f(x) has compact support. Second we studied that the f(x) has no compact support. Then we considered the singular limit of changing sign solutions of the porous medium equation. Subsequently, we study the porous medium equation with absorption and the very fast diffusion equation.The study of source type solutions of porous medium equations began at 1986. C. M. Elliott, M. herrero, J. R. King and J. R. Ockendon[1] considered the following Cauchy problem firstlyThe authors showed the properties of solutions as 0≤f(x)∈L^∞(R^N), m→∞.Caffarelli and A. Friedman [2] proved the convergence of solutions in L^∞(G), G (?) R^N×(0,∞) for problem (1.1)-(1.2), as 0≤f(x)∈L¹(R^N),m→∞.In 1998, KinMing Hui[5] considered the following Cauchy problem he proved that as 0≤f(x)∈L¹(R^N)∩L^∞(R^N),m→∞the solutions of problem (1.3)-(1.4) converges in L_loc¹ to the solutions of the ODE. KinMing Hui[6] also considered the above problem as p→∞.The study of source type solutions of p-Laplace equations by Evans and Feldman [7], they consideredThey proved that when the initial value f(x) has compact support, the solutions of problem (1.5)-(1.6) converges. The initial value f(x) has no compact support by Yi and Zhao [9].In this thesis, we introduce the following results and the proof.定理1 Let f(x)∈L^∞(R^N), |(?)f|∈L^∞(R^N). Then there exists a subsequence {u_Pj} of solutions {u_p} for the problem (??)-(??) and a function u_∞∈C(R^N),||(?)u_∞||_L∞≤1,such that for any compact set G (?) Qt,定理2 Let f(x)∈L¹(R^N),f(x)≥0,|(?)f|∈L^∞(R^N). Then there exists a function u_∞∈C(R^N),|(?)u_∞|≤1 such that for any compact set G (?) Q_t,and 定理3 The problem (??)-(??) has a unique solutionand...