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| THE FIRST BOUNDARY VALUE PROBLEM FOR QUASILINEAR DEGENERATE PARABOLIC EQUATIONS OF SECOND ORDER IN SEVERALSPACE VARIABLES |
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Citation： |
WuZhouqun,Zhao Junning.THE FIRST BOUNDARY VALUE PROBLEM FOR QUASILINEAR DEGENERATE PARABOLIC EQUATIONS OF SECOND ORDER IN SEVERALSPACE VARIABLES[J].Chinese Annals of Mathematics B,1983,4(1):57~76 |
| Page view： 689
Net amount： 729 |
Authors： |
WuZhouqun; Zhao Junning |
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| Abstract： |
In this paper， we study the first boundary value problem for quasilinear equations of the form
$$\[Lu \equiv \frac{{\partial u}}{{\partial t}} - \sum\limits_{i,j = 1}^m {\frac{\partial }{{\partial {x_i}}}({a^{ij}}(u,x,t)\frac{{\partial u}}{{\partial {x_j}}}) - \sum\limits_{i = 1}^m {\frac{\partial }{{\partial {x_i}}}{f^i}(u,x,t) = g(u,x,t)} } \]$$
with $\[{a^{ij}} = {a^{ji}}\]$ and
$$\[\sum\limits_{i,j = 1}^m {{a^{ij}}(u,x,t){\xi _i}{\xi _j}} \ge 0,\forall \xi = ({\xi _1}, \cdots {\xi _m}) \in {R^m}\]$$
Under certain conditions, the existence of generalized solutions in BV is proved by means of the method of parabolic regularization. To do this, we need some estimates on the family $\[\left\{ {{u_\varepsilon }} \right\}\]$ of solutions of regularized problems and the most difficult step is to estimate $\[{\left| {grad{u_\varepsilon }} \right|_{{L^1}}}\]$.
In addition, some results on the uniqueness and stability of generalized solutions are established. |
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