The nonlinear singular Burgers equation with small parameter and p-regularity theory

Abstract

In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation F(u,ε)=ut′-uxx′′+uux′+εu2=f(x,t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} F(u,\\varepsilon )=u_{t}'-u_{xx}''+uu_{x}'+\\varepsilon u^{2}=f(x,t), \\end{aligned}$$\\end{document}where F: Omega rightarrow mathcal {C}([0,pi ]times [0,infty )), Omega = mathcal {C}^{2}([0,pi ]times [0,infty ))times mathbb {R}, u(0,t)=u(pi ,t) =0, u(x,0)=g(x), and F, f(x, t), g(x) will be describe in the text. The first derivative of operator F at the solution point is degenerate. By virtue of p-regularity theory and Michael selection theorem, we prove the existence of continuous solution for this nonlinear problem.