Abstract
Consider the following anisotropic degenerate parabolic equation: $$\frac{\partial u}{\partial t} =\frac{\partial}{\partial x_{i}} \biggl(a^{ij}(u)\frac{\partial u}{\partial x_{j}} \biggr)+\frac{\partial b_{i}(u)}{\partial x_{i}},\quad (x,t)\in\Omega\times(0,T), $$ with the homogeneous Dirichlet boundary value. If the equation is not only degenerate in the interior of Ω, but also on the boundary ∂Ω, the paper discusses how to quote the suitable partly boundary condition to assure the well-posedness of an entropy solution of the equation. In particular, it is possible that the solution of the equation is free from the limitation of the boundary condition.
Highlights
The paper is to consider the anisotropic degenerate parabolic equation of the form+ ∂bi(u), ∂ xi in QT = × (, T), ( . )where ⊂ RN is an open bounded domain and the boundary ∂ = is C, is a symmetric matrix with nonnegative characteristic values, i.e., for any ξ ∈ RN, aij = aji, aijξiξj ≥, the pairs of the indices i, j imply the sum from to N
Zhan Boundary Value Problems (2015) 2015:22. It arises in the boundary layer theory, w wηη – wτ – ηUwξ + Awη + Bw =, ( . )
Where A, B are two known functions derived from the Prandtl system, one can refer to [ ] for details
Summary
If we want to consider the initial boundary value problem of equation Fichera [ , ] and Oleınik [ , ] developed and perfected the general theory of second order equation with a nonnegative characteristic form, which in particular contains those degenerating on the boundary. ), according to Fichera-Oleınik theory, the initial and the boundary value conditions for w have the form w|τ= = w (ξ , η), w|η= = , νwwη – v w + c(τ , ξ ) |η= = ,
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