Beata Medak

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.

The article discusses various modifications of the nonlinear Burgers equation with small parameter and degenerate in solution of the form\(F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} - f(x,t) = 0,\) (1)where \(F:\Omega \to C([0,\pi ] \times [0,T])\), \(T 0\), \(\Omega = {{C}^{2}}([0,\pi ] \times [0,T]\,)\,\mathbb{R}\) 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 ε with oscillating initial conditions of the form \(\varphi (x) = k\sin x\), where k –some, generally speaking, constant depending on ε, and study the question of the existence of a solution in neighborhood of the trivial \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), which corresponds to \(k = k{\kern 1pt} * = 0\) 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 ε.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 \(v(t) = c{{e}^{{ - t}}}\), \(u(x) \in {{\mathcal{C}}^{2}}([0,\pi ])\). In this case, the problem under consideration is degenerate at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), since \({\text{Im}}F_{u}^{'}(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) \ne Z = \mathcal{C}([0,\pi ] \times [0,T])\). 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 \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), т.е. p = 3.

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.