On Solution Existence for a Singular Nonlinear Burgers Equation with Small Parameter and p-Regularity Theory

Abstract

In this paper we study a solution existence problem for the singular nonlinear Burgers equation $F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} = f(x,t),$ where $F:\Omega \to C([0,\pi ] \times [0,T])$ and $\Omega = {{C}^{2}}([0,\pi ] \times [0,T]) \times \mathbb{R}$, with small parameter ε under the boundary conditions $u(0,t) = u(\pi ,t) = 0$ and the oscillating initial condition $u(x,0) = k\sin x.$ The first derivative of the operator F at the solution point is assumed to be degenerate, i.e., $F{\kern 1pt} '(x{\kern 1pt} \text{*})$ is not surjective. The existence of a continuous solution to this nonlinear problem is proved by applying p-regularity theory and the Michael selection theorem.