| Li DAQIAN,YU WENCHI,SEEN WEIXI.[J].数学年刊A辑,1981,2(1):65~90 |
|
| SECOND INITIAL-BOUNDARY VALUE PROBLEMS FORQUASI-LINEAR HYPERBOLIC-PARABOLICCOUPLED SYSTEMS |
| Received:December 25, 1979 |
| DOI： |
| 中文关键词: |
| 英文关键词: |
| 基金项目: |
|
| Hits: 498 |
| Download times: 0 |
| 中文摘要: |
| |
| 英文摘要: |
| On a rectangular domain
\[R(\delta ) = \{ 0 \leqslant t \leqslant \delta ,0 \leqslant x \leqslant 1\} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\]
We consider the second initial-boundary value problem for the quasi-linear hyperbolic- parabolic coupled system
\[{\begin{array}{*{20}{c}}
{\sum\limits_{j = 1}^n {{\zeta _{ij}}(t,x,u,v)(\frac{{\partial {u_j}}}{{\partial t}} + {\lambda _l}(t,x,u,v,{v_x})\frac{{\partial {u_j}}}{{\partial x}})} } \\
{ = {\zeta _l}(t,x,u,v)(\frac{{\partial v}}{{\partial t}} + {\lambda _l}(t,x,u,v,{v_x})\frac{{\partial v}}{{\partial x}})} \\
{ + {\mu _l}(t,x,u,v,{v_x}),(l = 1,...,n)} \\
{\frac{{\partial v}}{{\partial t}} - a(t,x,u,v,{v_x})\frac{{{\partial ^2}v}}{{\partial {x^2}}} = b(t,x,u,v,{v_x})}
\end{array}}\]
without loss of generatity,the initial conditions may be written as
\[t = 0,{u_j} = 0,(j = 1,...,n),v = 0\]
and we can suppose that
\[\left\{ {\begin{array}{*{20}{c}}
{a(0,x,0,0,0) \equiv 1} \\
{b(0,x,0,0,0) \equiv 0} \\
{{\zeta _{ij}}(0,x,0,0) \equiv {\delta _{lj}} = \left\{ {\begin{array}{*{20}{c}}
{1,if{\kern 1pt} {\kern 1pt} {\kern 1pt} l = j} \\
{0,if{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \ne j}
\end{array}} \right.}
\end{array}} \right.\]
The boundary conditions are as follows:
\[\begin{gathered}
on{\kern 1pt} {\kern 1pt} {\kern 1pt} x = 1,\left\{ {\begin{array}{*{20}{c}}
{{u_{\bar r}} = {G_{\bar r}}(t,u,v),(\bar r = 1,...,h;h \leqslant n)} \\
{\frac{{\partial v}}{{\partial x}} = {F_ + }(t,u,v);}
\end{array}} \right. \hfill \ on{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{c}}
{{u_{\hat s}} = {{\hat G}_{\hat s}}(t,u,v),(\hat s = m + 1,...,n;m \geqslant 0)} \\
{\frac{{\partial v}}{{\partial x}} = {F_ - }(t,u,v){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} }
\end{array}} \right. \hfill \\
\end{gathered} \]
Uf = Q-f(t> u, x), (r = 1> k^n),
We assume that the following conditions are satisfied:
(1) the orientability condition
\[\begin{gathered}
{\lambda _{\bar r}}(0,1,0,0,0) < 0,{\lambda _s}(0,1,0,0,0) > 0,\left( {\begin{array}{*{20}{c}}
{\bar r = 1,...,h} \\
{s = h + 1,...,n}
\end{array}} \right) \hfill \ {\lambda _{\bar r}}(0,0,0,0,0) < 0,{\lambda _{\hat s}}(0,0,0,0,0) > 0,\left( {\begin{array}{*{20}{c}}
{\hat r = 1,...,m} \\
{\hat s = m + 1,...,n}
\end{array}} \right) \hfill \\
\end{gathered} \]
(2) the compatibility condition
\[\begin{gathered}
\frac{{\partial {G_{\bar r}}}}{{\partial t}}(0,0,0) + \sum\limits_{j = 1}^n {\frac{{\partial {G_{\bar r}}}}{{\partial {u_j}}}} (0,0,0){\mu _j}(0,1,0,0,0) = {\mu _{\bar r}}(0,1,0,0,0) \hfill \ \frac{{\partial {{\hat G}_{\hat s}}}}{{\partial t}}(0,0,0) + \sum\limits_{j = 1}^n {\frac{{\partial {{\hat G}_{\hat s}}}}{{\partial {u_j}}}} (0,0,0){\mu _j}(0,0,0,0,0) = {\mu _{\hat s}}(0,0,0,0,0) \hfill \ (\bar r = 1,...,h;\hat s = m + 1,...,n);{F_ \pm }(0,0,0) = 0 \hfill \\
\end{gathered} \]
(3) the condition of characterizing number
\[\begin{gathered}
\sum\limits_{j = 1}^n {\left| {\frac{{\partial {G_{\bar r}}}}{{\partial {u_j}}}(0,0,0)} \right|} < 1 \hfill \ \sum\limits_{j = 1}^n {\left| {\frac{{\partial {{\hat G}_{\hat s}}}}{{\partial {u_j}}}(0,0,0)} \right|} < 1(\bar r = 1,...,h,\hat s = m + 1,...,n \hfill \\
\end{gathered} \]
(4)The smoothness condition: the coefficients of the system and the boundary conditions are suitably smooth.
By means of certain a priori estimations for the solution of the heat equation and the linear hyperbolic system, using an iteration method and Leray-Schauder fixed point theorem, we have proved
Theorem 1. Under the preceding hypotheses, for the second initial-boundary value problem (2)—(4), (6), (7), there exists uniquely a classical solution on R(8) where \[\delta \]>0 is suitably small.
Theorem 2. In theorem the 1,condition of characterizing number (13) may be ameliorated as the following solvable condition;
\[\left\{ {\begin{array}{*{20}{c}}
{\det |({\delta _{\bar rr'}} - \frac{{\partial {G_{\bar r}}}}{{\partial {u_{r'}}}}(0,0,0)| \ne 0,(\bar r,r' = 1,...,h)} \\
{\det |({\delta _{\hat s\hat s'}} - \frac{{\partial {G_{\hat s}}}}{{\partial {u_{\hat s'}}}}(0,0,0)| \ne 0,(\hat s,\hat s' = m + 1,...,n)}
\end{array}} \right.\]
i.e,the boundary condition (6),(7)may be written as
\[\begin{gathered}
on{\kern 1pt} {\kern 1pt} {\kern 1pt} x = 1,\left\{ {\begin{array}{*{20}{c}}
{{u_{\bar r}} = {H_{\bar r}}(t,{u_s},v),} \\
{\frac{{\partial v}}{{\partial x}} = {F_ + }(t,u,v);}
\end{array}} \right.\left( {\begin{array}{*{20}{c}}
{\bar r = 1,...,h} \\
{s = h + 1,...,n}
\end{array}} \right) \hfill \ on{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{c}}
{{u_{\hat s}} = {H_{\hat s}}(t,{u_{\hat r}},v){\kern 1pt} ,} \\
{\frac{{\partial v}}{{\partial x}} = {F_ - }(t,u,v){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} }
\end{array}} \right.\left( {\begin{array}{*{20}{c}}
{\hat r = 1,...,m} \\
{\hat s = m + 1,...,n}
\end{array}} \right) \hfill \\
\end{gathered} \] |
| View Full Text View/Add Comment Download reader |
| Close |
|
|
|
|
|
