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.

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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.

  • History

Published online: 1996-09

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