THE GLOBAL SOLUTION OF INITIALBOUNDARY VALUE PROBLEM OF HIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE

In practical problems there appears the higher-order equations of changing type. But,there is only a few of papers, which studied the problems for this kind of equations. In this paper a kind of the higher-order m

THE GLOBAL SOLUTION OF IN ITIAL BOUNDARY VALUE PROBLEM OF HIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE

In practical problems there appears the higher-order equations of changing type. But, there is only a few of papers, which studied the problems for this kind of equations. In this paper a kind of the higher-order multidimensional equations of changing type is considered: $$\Lu \equiv k(t){u_{tt}} + {( - 1)^{M - 1}}\sum\limits_{|\alpha |,|\beta | = M} {D_x^\alpha ({a_{\alpha \beta }}(x)D_x^\beta u)} - b(x,t){u_t} = f(x,t),\]$$ where $\M \ge 1\]$,$\x = ({x_1}, \cdots ,{x_n}) \in \Omega \subset {R^n},t \in 0,T],k(t) > 0\]$ when $\t = 0,k(t) \ge 0\]$ when $\t \in (0,{t_0}),k(t) = 0\]$ when $\t = {t_0}\]$ and $\k(t) \le 0\]$ when $\t \in ({t_0},T]\]$. The existence and uniqueness of the global regular solution of the initial-boundary value problem $$\\left\{ {\begin{array}{*{20}{c}} {{D^\upsilon }u = 0,0 \le |\upsilon | \le M - 1on\partial \Omega \times 0,T]}\{u(x,0) = 0,x \in \Omega } \end{array}} \right.\]$$ for this equation are proved. Moreover, the result is generalized to a semi-linear equation.