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| INITIAL VALUE PROBLEMS FOR NONLINEAR DEGENERATE SYSTEMS OF FILTRATION TYPE |
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Citation： |
Zhou Yulin.INITIAL VALUE PROBLEMS FOR NONLINEAR DEGENERATE SYSTEMS OF FILTRATION TYPE[J].Chinese Annals of Mathematics B,1984,5(4):633~652 |
| Page view： 651
Net amount： 731 |
Authors： |
Zhou Yulin; |
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| Abstract： |
In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type $\[{u_t} = (grad\varphi (u))\]$ are studied, where $\[u = ({u_1}, \cdots ,{u_N})\]$ is an N-dimensional vector valued function, $\[\varphi (u)\]$ is a strict convex function of vector variable $\[u\]$, and its matrix of derivatives of second order is zero-definite at $\[u = 0\]$. This system is degenerate. The definition of the generalized solution of the problem: $\[u(x,t) \in {L_\infty }((0,T);{L_2}(R)),\]$, grad $\[\varphi (u) \in {L_\infty }((0,T);W_2^{(1)}(R)),\]$ and it satisfies appropriate integral relation. The existence and uniqueness of the generalized solution of the problem are proved. When N=1, the system is the commonly so-called degenerate partial differential equation of filtration type. |
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