To the question of global solvability of mixed problems for quasilinear parabolic equations
The following initial-boundary value problem is studied: \[ \begin{gathered} u_t = a(x,u,u_x)u_{xx} + b(x,u,u_x), \\ u|_{x=0} = \xi_0, \quad u|_{x=1} = \xi_1,\quad u|_{t=0} = u_0(x). \end{gathered} \] It is assumed that \(a(x,\xi,\eta), b(x,\xi,\eta) \in C^{\infty}\) for \(0\leq x\leq 1\), \(-\infty <\xi<\infty\), \(-\infty <\eta<\infty\); \(a(x,\xi,\eta)\geq \delta >0\), with \(\delta\) a constant; \(u_0(x)\in C^2\) for \(0\leq x\leq 1\); \(u_0(0)=\xi_0\); and \(u_0(1)=\xi_1\). The authors observe that there exists a close connection between solutions to the corresponding variational problem and solutions to the nonstationary problem. In the present article the minimizing problem for the functional \[ I(x_1,x_2,y(x)) = \int\limits_{x_1}^{x_2}F(x,y') dx \] is studied. Here \(0\leq x_1<x\leq x_2\leq 1\) and \(F(x,y') = y^{\prime 2}/(1 + \sqrt{1+x^2 y^{\prime 2}})\). A minimal value of the functional is searched among the functions \(y(x)\in AC^*\) (the class of absolutely continuous functions \(y(x)\) such that \(|y'(x)|\leq K_y\) for almost all \(x\in [x_1,x_2]\), with \(K_y\) a constant) satisfying the conditions \(y(x_1) = y_1,\;y(x_2) = y_2\), where \(0\leq y_1<y_2\). Properties of minimizing sequences for the functional are elaborated.
scientific article; zbMATH DE number 1431667