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| THE GLOBAL SOLUTION OF IN ITIAL BOUNDARY VALUE PROBLEM OFHIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE |
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Citation： |
Sun Hesheng.THE GLOBAL SOLUTION OF IN ITIAL BOUNDARY VALUE PROBLEM OFHIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE[J].Chinese Annals of Mathematics B,1988,9(4):429~435 |
| Page view： 838
Net amount： 774 |
Authors： |
Sun Hesheng; |
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| Abstract： |
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. |
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