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| THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS |
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Citation： |
Qin Tiehu.THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS[J].Chinese Annals of Mathematics B,1988,9(3):251~269 |
| Page view： 790
Net amount： 762 |
Authors： |
Qin Tiehu; |
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| Abstract： |
The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:
$$\[{u_{tt}} - \sum\limits_{i,j = 1}^n {{a_{ij}}(Du){u_{{x_i}{x_j}}}} = 0,in(0,\infty ) \times \Omega ,\]$$
$$\[u{|_{{\Gamma _0}}} = 0\]$$
$$\[\sum\limits_{i,j = 1}^n {{a_{ij}}(Du){n_j}{u_{{x_i}}} + b(Du){u_i}{|_{{\Gamma _i}}}} = 0,\]$$
$$\[u{|_{t = 0}} = \varphi (x),{u_t}{|_{t = 0}} = \psi (x),in\Omega \]$$
where $\[\partial \Omega = {\Gamma _0} \cup {\Gamma _1},b(Du) \ge {b_0} > 0\]$. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data. |
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