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

In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation F ( u , ε ) = u t ′ - u xx ′ ′ + u u x ′ + ε u 2 = f ( x , t ) , where F : Ω → C ( [ 0 , π ] × [ 0 , ∞ ) ) , Ω = C 2 ( [ 0 , π ] × [ 0 , ∞ ) ) × R , u ( 0 , t ) = u ( π , 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.