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 $$\begin{aligned} F(u,\varepsilon )=u_{t}'-u_{xx}''+uu_{x}'+\varepsilon u^{2}=f(x,t), \end{aligned}$$ F ( u , ε ) = u t ′ - u xx ′ ′ + u u x ′ + ε u 2 = f ( x , t ) , where $$F: \Omega \rightarrow \mathcal {C}([0,\pi ]\times [0,\infty ))$$ F : Ω → C ( [ 0 , π ] × [ 0 , ∞ ) ) , $$\Omega = \mathcal {C}^{2}([0,\pi ]\times [0,\infty ))\times \mathbb {R}$$ Ω = C 2 ( [ 0 , π ] × [ 0 , ∞ ) ) × R , $$u(0,t)=u(\pi ,t) =0$$ u ( 0 , t ) = u ( π , t ) = 0 , $$u(x,0)=g(x)$$ 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.