表題番号：2009B-065
日付：2010/04/02

研究課題非線形放物ー双曲型方程式のエントロピー解の構造研究とその応用

研究者所属（当時） | 資格 | 氏名 | |
---|---|---|---|

（代表者） | 教育・総合科学学術院 | 教授 | 小林 和夫 |

- 研究成果概要

We proved the uniqueness of entropy solutions of non- isotropic degenerate parabolic- hyperbolic equations with non homogeneous Dirichlet boundary value condition.

Let Ω be an open bounded cube of Rｄ, with the boundary ∂Ω. Let QT = (0,T)×Ω, T> 0 and ΣT = (0,T)× ∂Ω. We consider

the following problem (P):

∂t u + ∑i=1 ∂Ai (u) /xi – Σi,j=1 ∂2 βij (u)/∂xi∂xj = g in QT,

u = ub on ΣT,

u(0, ) = u0 in Ω,

where u(t,x): QT → R $, is the unknown function and ub :ΣT → R ,g: QT → R are given functions. We assume that

Ai(u): R → R is locally Lipschiz,

the d×d matrix (β’ ij(u))) is symmetric, nonnegative and locally bounded.

We have to explain the meaning of the boundary condition. In the nondegenerate case ( in which (β’ij(u)) is strictly positive) u = ub a.e. (t,x) in ΣT in the classical sense. In this case the problem (P) is of parabolic equation. However, the situation drastically changes for the completely degenerate case (in which β’ij(u) = 0 ). In this case

the problem (P) becomes a first order hyperbolic equation . It is well known that discontinuous solution must be considerd and discontinuous weaqk solutions are not uniquely determined by their data.

In this study we consider entropy solutions to the problem (P) and obtain the comparison theorem, which in particular deduces

the uniqueness of entropy solutions of (P). The precise results will appear in the forthcoming paper.