Internal layers of parabolic singularly perturbed problems



The aim of this work is to show, through analyzing some incorrect results in the literature, (i) the internal layer structure is not so predictable for the linear parabolic differential equations with constant coefficients; and (ii) the location of the viscous shock of Burgers' equation in the quarter plane depends on the viscosity, but not in the half plane. Our methods are based on asymptotics of integrals together with special functions to unveil these distinguished features. Contrary to the exponential function as the sole building block of boundary layer functions for differential equations and partial differential equations near the outflow boundary, the class of the complementary error function and its iterated integrals becomes fundamental in this investigation.