On the application of the solution of the degenerate nonlinear Burgers equation with a small parameter and the theory of $p$-regularity

The article discusses various modifications of the nonlinear Burgers equation with small parameter and degenerate in solution of the form
$$ F(u,\varepsilon)\equiv u_t=u_{xx}+uu_x+\varepsilon u^2-f(x,t)=0,\qquad (1) $$
where $F\colon \Omega\to C([0,\pi]\times [0,T])$, $T>0$, $\Omega=C^2([0,\pi]\times[0,T])\mathbb R$ and $u(0,t)=u(\pi,t)=0$, $u(x,0)=\varphi(x)$, $f(x,t)\in C([0,\pi]\times[0,T])$, $\varphi(x)\in C[0,\pi]$. We will be interested in the most important in applications case of a small parameter $\varepsilon$ with oscillating initial conditions of the form $\varphi(x)=k\sin{x}$, where $k$ – some, generally speaking, constant depending on $\varepsilon$, and study the question of the existence of a solution in neighborhood of the trivial $(u*,\varepsilon*)=(0,0)$, which corresponds to $k=k*=0$ and at what initial under certain conditions on the values of $k$, it is possible to construct an analytical approximation of this solution for small $\varepsilon$. We will look for a solution in the traditional way of separation of variables on a subspace of functions of the form $u(x,t)=v(t)u(x)$, where $v(t)=ce^{-t}$, $u(x)\in C^2([0,\pi])$. In this case, the problem under consideration is degenerate at the point $(u*,\varepsilon*)=(0,0)$, since $\operatorname{Im} F'_u(u*,\varepsilon*)\neq Z=C([0,\pi]\times[0,T])$. This follows from the Sturm–Liouville theory. To achieve our goals, we apply the apparatus of $p$-regularity theory [6, 7, 15, 16] and show that the mapping $F(u,\varepsilon)$ is $3$-regular at the point $(u*,\varepsilon*)=(0,0)$, т.е. $p=3$.