Existence of solution to Korteweg-de Vries equation in a non-parabolic domain (Q2309331)

In this work, the non-homogeneous Korteweg-de Vries problem (KdV) \[ \left \{\begin{array}{c}\partial _{t}u+c(t)u.\partial _{x}u+\partial _{x}^{3}u=f,\text{ \ \ in }\Omega , \\ u(t,\varphi _{1}(t))=u(t,\varphi _{2}(t))=\partial _{x}u(t,\varphi _{2}(t)=0,\text{ \ }t\in (0,T),\end{array}\right.\tag{\(\ast\)} \] is considered in the non parabolic bounded domain \[ \Omega =\{(t,x)\in \mathbb{R}^{2}:~\varphi _{1}(t)<x<\varphi _{2}(t),~0<t<T<+\infty\}. \] The coefficient \(c\) is known and the function \(f\) \(\ \) is given in Lebesgue space \(L^{2}(\Omega ).\) The same problem has been treated by \textit{Y. Benia} and \textit{B.-K. Sadallah} [Math. Methods Appl. Sci. 41, No. 7, 2684--2698 (2018, Zbl 06876782)] when \(\varphi_{1}(0)<\varphi _{2}(0).\) But here, the authors replace this condition by the singular one \(\varphi _{1}(0)=\varphi _{2}(0),\) and look for the solution \(u\) in the Sobolev space \(H^{1,3}(\Omega )\) defined by \[ H^{1,3}(\Omega )=\left \{ u\in L^{2}(\Omega ),\text{ }\partial _{t}u\in L^{2}(\Omega ),\text{ }\partial _{x}^{j}u\in L^{2}(\Omega ),\text{ }j=1,2,3\right \}. \] For this end, some other conditions on \(f\), \(c\), \(\varphi _{1}\) and \(\varphi _{2}\) are assumed, these are : \(\partial _{t}f\in L^{2}(\Omega ),\) and the functions \(c\), \((\varphi _{i})_{i=1,2},\) \((\varphi _{i}^{\prime})_{i=1,2},\) \((\varphi _{i}^{\prime \prime })_{i=1,2}\) are bounded. Then, the main result of the paper claims that Problem \((\ast )\) admits a unique solution \(u\in H^{1,3}(\Omega ).\) In the first step, the authors show some estimates for the solutions in subdomains \((\Omega_{n})_{n\in N}\) that can be transformed into a rectangle (i.e., the case when \(\varphi _{1}(0)<\varphi _{2}(0)).\)The second step uses the previous estimates to ``pass to the limit'' in order to obtain the ``triangular'' case (\(\varphi _{1}(0)=\varphi _{2}(0)\)). The method consists to approximate the triangular domain \(\Omega\) by a sequence of the subdomains \(\Omega_{n})_{n\in N}\). Observe that an essential part of the proof is based on the parabolic regularization method, and Faedo-Galerkin method.