Abstract
The degenerate parabolic equations from the reaction–diffusion problems are considered on an unbounded domain varOmegasubsetmathbb {R}^{N}. It is expected that only a partial boundary should be imposed the homogeneous boundary value, but how to give the analytic expression of this partial boundary seems very difficult. A new method, which is called the general characteristic function method, is introduced in this paper. By this new method, a reasonable analytic expression of the partial boundary value condition is found. Moreover, the stability of the entropy solutions is established based on this partial boundary value condition.
Highlights
In the theory of water infiltration through porous media, Darcy’s linear relationV = –K(θ )∇φ, satisfactorily describes the flow conduction provided that the velocities are small, where V represents the seepage velocity of water, θ is the volumetric moisture content, K(θ ) is the hydraulic conductivity and φ is the total potential, which can be expressed as the sum of a hydrostatic potential ψ(θ ) and a gravitational potential z φ = ψ(θ) + z.But when the flow has large velocities, Darcy’s linear relation is invalid
No boundary value being imposed on ∂Ω \ Σp implies that there is a thermal insulation on ∂Ω \ Σp, the heat conduction cannot pass ∂Ω \ Σp
For a parabolic–hyperbolic equation, how to impose a suitable partial boundary value condition to ensure the well-posedness of the entropy solutions is a very interesting problem
Summary
In the theory of water infiltration through porous media, Darcy’s linear relation. V = –K(θ )∇φ, satisfactorily describes the flow conduction provided that the velocities are small, where V represents the seepage velocity of water, θ is the volumetric moisture content, K(θ ) is the hydraulic conductivity and φ is the total potential, which can be expressed as the sum of a hydrostatic potential ψ(θ ) and a gravitational potential z φ = ψ(θ) + z. When the flow has large velocities, Darcy’s linear relation is invalid. In this case, in order to obtain a more accurate description of the flow, several nonlinear versions have been proposed. In order to obtain a more accurate description of the flow, several nonlinear versions have been proposed One of these versions is V = –K(θ )∇φ.
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