### Abstract

The existence of strong solutions of Cauchy problem for the following evolution equation du(t)/dt + ∂^{1} (u (t)) - ∂^{2} (u(t)) ∋ f(t) is considered in a real reflexive Banach space V, where ∂^{1} and ∂^{2} are subdifferential operators from V into its dual V*. The study for this type of problems has been done by several authors in the Hilbert space setting. The scope of our study is extended to the V-V* setting. The main tool employed here is a certain approximation argument in a Hilbert space and for this purpose we need to assume that there exists a Hilbert space and for this purpose we need to assume that there exists a Hilbert space H such that V ⊂ H ≡ H* ⊂ V* with densely defined continuous injections. The applicability of our abstract framework will be exemplified in discussing the existence of solutions for the nonlinear heat equation: u_{t}(x, t) - Δ_{p}u(x,t)- u ^{q-2}u(x,t) = f(x,t), x ∈ Ω t > 0, u/∂Ω = 0, where Ω is a bounded domain in ℝ^{N}. In particular, the existence of local (in time) weak solution is shown under the subcritical growth condition q < p* (Sobolev's critical exponent) for all data u_{0} ∈ W_{0}
^{1,p} (Ω) This fact has been conjectured but left as an open problem through many years.

Original language | English |
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Pages (from-to) | 392-415 |

Number of pages | 24 |

Journal | Journal of Differential Equations |

Volume | 209 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Feb 15 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis

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## Cite this

*Journal of Differential Equations*,

*209*(2), 392-415. https://doi.org/10.1016/j.jde.2004.08.011