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| PROPAGATION OF SINGULARITES FOR SOLUTION OF SEMILINER WAVE EQUATIONS |
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Citation： |
Liu Linqi.PROPAGATION OF SINGULARITES FOR SOLUTION OF SEMILINER WAVE EQUATIONS[J].Chinese Annals of Mathematics B,1988,9(4):379~389 |
| Page view： 834
Net amount： 786 |
Authors： |
Liu Linqi; |
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| Abstract： |
In order better to research the singularities of the solutions $\[u \in H_{loc}^s(\Omega ),\Omega \subset {R^n},s > \frac{n}{2} + 1\]$ , for semilinear hyperbolic equations $\[u = f(u,Du)\]$, in this paper, a kind of weighted Sobolev space $\[({H^s})_{{P_\mu }}^\alpha \],\[\mu = 1,2,{p_1} = {D_i} - \left| {{D_x}} \right|,{P_2} = {D_i} + \left| {{D_x}} \right|\]$, closely related with the solutions of the equations, is presented. It is discussed that their products tacitly keep roughly $\[{H^{3x - n}}\]$ microlocal regularity on the characteristic directions for $\[{P_\mu }\]$ and invariance under nonlinear maps. Then it is obtained that roughly $\[{H^{3x - n}}\]$ propagation of singularities theorem is valid for $\[u = f(u)\]$. |
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