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Volume 9, Issue 2
On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

Ning Zhu

J. Part. Diff. Eq., 9 (1996), pp. 129-138.

Published online: 1996-09

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  • 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.
  • Keywords

Filtration type Cauchy problem initial trace existence of solutions

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@Article{JPDE-9-129, author = {Ning Zhu }, 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 AU - Ning Zhu 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 . (1996). On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity. Journal of Partial Differential Equations. 9 (2). 129-138. doi:
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