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On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

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@Article{JPDE-9-129,
author = {},
title = {On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity},
journal = {Journal of Partial Differential Equations},
year = {1996},
volume = {9},
number = {2},
pages = {129--138},
abstract = { In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.},
issn = {2079-732X},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jpde/5615.html}
}

TY - JOUR
T1 - On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity
JO - Journal of Partial Differential Equations
VL - 2
SP - 129
EP - 138
PY - 1996
DA - 1996/09
SN - 9
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5615.html
KW - Filtration type
KW - Cauchy problem
KW - initial trace
KW - existence of solutions
AB - In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.

Ning Zhu . (2019). On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity.

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*Journal of Partial Differential Equations*.*9*(2). 129-138. doi: