Asymptotics of the gradient of the solution and the inception of singularities in the initial-boundary problem for an equation of a composite type
On the plane of variables \(x=(x_1,x_2) \in\mathbb{R}^2\), consider the contour \( \Gamma\) that consists of \(N\) slits \(\Gamma_1, \dots, \Gamma_N\) located along the axis \(Os\) obtained from the axis \(Ox_1\) by the rotation about the polar angle \(\theta\) around the origin: \[ \Gamma= \bigcup^N_{n=1} \Gamma_n, \quad \Gamma_n= \bigl\{x:x_1 =s\cos \theta,\;x_2= s \sin \theta,\;s\in(a_n,b_n) \bigr\}. \] We assume that the slits \(\Gamma_n\) have no common ends. We retain the notations \(a_n\) and \(b_n\) for the points \((a_n\cos \theta\), \(a_n \sin \theta) \) and \((b_n\cos \theta\), \(b_n\sin\theta)\) on the plane \((x_1,x_2)\) and denote by \(X\) the set of points of the plane that consists of the ends of the contour \(\Gamma\): \(X= \bigcup^N_{n=1} (a_n \cup b_n)\). We denote by \(\Gamma^+\) the side of the contour \(\Gamma\) that remains on the left as the parameter \(s\) increases; the opposite side will be denoted by \(\Gamma^-\). Consider the initial-boundary problem for an equation of a composite type: Find a function \(u(t,x)\) from the smoothness class \(G\) that satisfies (in the classical sense) the equation \[ {\partial^2 \over \partial t^2} \Delta u+ \omega^2_1 u_{x_1x_1} +\omega^2_2 u_{x_2x_2} =0, \] in \((0,\infty) \times\mathbb{R}^2 \setminus \Gamma\), the initial conditions \(u(0,x)= u_t(0,x) =0\), \(x\in\mathbb{R}^2 \setminus \Gamma\), the boundary conditions \(u|_{x(s) \in\Gamma^+} =f^+(t,s)\), \(u |_{x(s) \in\Gamma^-} =f^-(t,s)\), and the regularity conditions at infinity \[ | D^k_tu |\leq B_k(t); \quad | D^k_t \nabla u| \leq\overline B_k(t) | x|^{-1 -\varepsilon}, \quad k=0, 1,2, \] where \(B_k(t)\), \(\overline B_k(t)\in C^0 [0,\infty)\), and \(\varepsilon>0\). We denote this solution by \(u(t,x)\). Our purpose is to study the behavior of the derivatives \(u_{x_1} (t,x)\) and \(u_{x_2} (t,x)\) for large values of time.
scientific article; zbMATH DE number 1155312