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Volume 10, Issue 1
Hyperbolic Phenomena in a Degenerate Parabolic Equation

Fuxia Cheng

J. Part. Diff. Eq., 10 (1997), pp. 85-96.

Published online: 1997-10

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  • Abstract
M. Bertsch and R. Dal Passo [1] considered the equation u_t =  (φ(u)ψ(u_z))x., where φ > 0 and ψ is a strictly increasing function with lim_{s → ∞} ψ(s) = ψ_∞ < ∞. They have solved the associated Cauchy problem for an increasing initial function. Furthermore, they discussed to what extent the solution behaves like the solution of the first order conservation law u_t = ψ_∞(φ(u))_x. The condition φ > 0 is essential in their paper. In the present paper, we study the above equation under the degenerate condition φ(0) = 0. The solution also possesses some hyperbolic phenomena like those pointed out in [1].
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@Article{JPDE-10-85, author = {}, title = {Hyperbolic Phenomena in a Degenerate Parabolic Equation}, journal = {Journal of Partial Differential Equations}, year = {1997}, volume = {10}, number = {1}, pages = {85--96}, abstract = { M. Bertsch and R. Dal Passo [1] considered the equation u_t =  (φ(u)ψ(u_z))x., where φ > 0 and ψ is a strictly increasing function with lim_{s → ∞} ψ(s) = ψ_∞ < ∞. They have solved the associated Cauchy problem for an increasing initial function. Furthermore, they discussed to what extent the solution behaves like the solution of the first order conservation law u_t = ψ_∞(φ(u))_x. The condition φ > 0 is essential in their paper. In the present paper, we study the above equation under the degenerate condition φ(0) = 0. The solution also possesses some hyperbolic phenomena like those pointed out in [1].}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5583.html} }
TY - JOUR T1 - Hyperbolic Phenomena in a Degenerate Parabolic Equation JO - Journal of Partial Differential Equations VL - 1 SP - 85 EP - 96 PY - 1997 DA - 1997/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5583.html KW - Degenerate parabolic equation KW - entropy condition AB - M. Bertsch and R. Dal Passo [1] considered the equation u_t =  (φ(u)ψ(u_z))x., where φ > 0 and ψ is a strictly increasing function with lim_{s → ∞} ψ(s) = ψ_∞ < ∞. They have solved the associated Cauchy problem for an increasing initial function. Furthermore, they discussed to what extent the solution behaves like the solution of the first order conservation law u_t = ψ_∞(φ(u))_x. The condition φ > 0 is essential in their paper. In the present paper, we study the above equation under the degenerate condition φ(0) = 0. The solution also possesses some hyperbolic phenomena like those pointed out in [1].
Fuxia Cheng . (2019). Hyperbolic Phenomena in a Degenerate Parabolic Equation. Journal of Partial Differential Equations. 10 (1). 85-96. doi:
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