## Abstract

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: u_{t} (x, t) - Δ_{p} u (x, t) - | u |^{q - 2} u (x, t) = f (x, t), (x, t) ∈ Ω × (0, T), where 2 ≤ p < q < + ∞, Ω is a bounded domain in R^{N}, f : Ω × (0, T) → R is given and Δ_{p} denotes the so-called p-Laplacian defined by Δ_{p} u : = ∇ ṡ (| ∇ u |^{p - 2} ∇ u), with initial data u_{0} ∈ L^{r} (Ω) is proved under r > N (q - p) / p without imposing any smallness on u_{0} and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, L^{r}-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0, T_{0}] in which the problem admits a solution. More precisely, T_{0} depends only on | u_{0} |_{Lr} and f.

Original language | English |
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Pages (from-to) | 359-385 |

Number of pages | 27 |

Journal | Journal of Differential Equations |

Volume | 241 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 Oct 15 |

Externally published | Yes |

## Keywords

- Degenerate parabolic equation
- Local existence
- Reflexive Banach space
- Subdifferential
- p-Laplacian

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics