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Volume 10, Issue 3
Regularity Results for a Strongly Degenerate Parabolic Equation

Fuxia Cheng

J. Part. Diff. Eq., 10 (1997), pp. 275-283.

Published online: 1997-10

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  • Abstract
M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.
  • Keywords

Strongly degenerate parabolic equation uniformly parabolic equation supersolution

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@Article{JPDE-10-275, author = {Fuxia Cheng }, title = {Regularity Results for a Strongly Degenerate Parabolic Equation}, journal = {Journal of Partial Differential Equations}, year = {1997}, volume = {10}, number = {3}, pages = {275--283}, abstract = { M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5597.html} }
TY - JOUR T1 - Regularity Results for a Strongly Degenerate Parabolic Equation AU - Fuxia Cheng JO - Journal of Partial Differential Equations VL - 3 SP - 275 EP - 283 PY - 1997 DA - 1997/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5597.html KW - Strongly degenerate parabolic equation KW - uniformly parabolic equation KW - supersolution AB - M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.
Fuxia Cheng . (1997). Regularity Results for a Strongly Degenerate Parabolic Equation. Journal of Partial Differential Equations. 10 (3). 275-283. doi:
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