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- W1974297325 abstract "In this paper we consider the differential properties of the solution to the Cauchy problem for the quasilinear parabolic equation [ (1)qquad frac{{partial v}}{{partial t}} = frac{1}{2}sumlimits_{i,j = 1}^n {c_{ij} } (t,x,v)frac{{partial ^2 v}}{{partial x_i partial x_i }} + sumlimits_{i = 1}^n {a_i } (t,x,v)frac{{partial v}}{{partial x_i }}, ] where $c_{ij} = sumnolimits_{k = 1}^n {b_{ik} } (t,x,v)b_{jk} (t,x,v)$. Let class $C_T^{m,gamma } $ be a class of continuous functions, which has bounded space derivatives up to the m-th order, and its m-th derivative is Hölder continuous with Hölder constant $gamma $ It is proved in this paper that if [ { {b(t,x,v),a(t,x,v),psi (x)} } in C_T^{m,gamma } ,qquad m geqq 2, ] then $v in C_{t_0 }^{m,gamma } $, where $t_0 > 0$ depends only on the constants of class $C_T^{m,gamma } $. If $n = 1$, $a(t,x,v) equiv 0$, then the above assertion will follow for all $t in [0,T]$ and $x in ( - infty ,infty )$. It is noted that $b(t,x,v)$ may be any degenerate matrix. These assertions are proved by the method of diffusion processes." @default.
- W1974297325 created "2016-06-24" @default.
- W1974297325 creator A5040793403 @default.
- W1974297325 date "1964-01-01" @default.
- W1974297325 modified "2023-10-18" @default.
- W1974297325 title "Cauchy’s Problem for Degenerate Quasilinear Parabolic Equations" @default.
- W1974297325 doi "https://doi.org/10.1137/1109050" @default.
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