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| ON FINITE DIFFUSING SPEED FOR UNIFORMLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS |
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Citation： |
Chen Yazhe.ON FINITE DIFFUSING SPEED FOR UNIFORMLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS[J].Chinese Annals of Mathematics B,1986,7(3):318~329 |
| Page view： 867
Net amount： 846 |
Authors： |
Chen Yazhe; |
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| Abstract： |
In this paper, the author studies the quasilinear parabolic equation
$$\[{u_t} = {({a_{ij}}(x,t,u){u_{{x_j}}})_i} + {b_i}(x,t,u){u_{{x_i}}} + c(x,t,u)\]$$
In $\[{Q^T} = \{ (x,t)|x \in {R^N},0 < t \le T\} \]$, which. is uniformly degenerate w herever $\[u = 0\]$. Under some conditions of the coefficients of the equation and the presupposition $\[0 \le u(x,t) \le M,\]$, the author proves that non-negative weak solutions, $\[u(x,t)\]$, to the equation satisfy the estimation that, for any $\[({x^0},{t^0}) \in {Q^T}\]$,
$$\[\frac{1}{C}\min \left\{ {\mathop {\inf }\limits_{|x - {x^0}| \le b\sqrt {{t^0}} } u(x,0),1} \right\} \le u({x^0},{t^0}) \le C\mathop {\sup }\limits_{|x - {x^0}| \le b\sqrt {{t^0}} } u(x,0),\]$$
where the constants b and C depend only upon M, T and the coefficients. It is a more exact description on the finite diffusing speed for the equation which has not been obtained even for one-dimensional porous medium e uations. |
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