### Abstract

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v^{1}(x), v^{2}(x),..., v^{m}(x)) from Ω ⊂ R^{n} into R^{m} which satisfy ψ^{i}(x, t) ≤ v^{i}(x) ≤ θ^{i}(x, t) for all x ε{lunate} Ω and 1 ≤ i ≤ m, where ψ^{i}and θ^{i} are extended real-valued functions on \ ̄gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u^{1}(x, t), u^{2}(x, t),..., u^{m}(x, t)) which satisfies U(x, 0) ε{lunate} K(0) will also satisfy U(x, t) ε{lunate} K(t) for all 0 ≤ t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.

Original language | English (US) |
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Pages (from-to) | 166-181 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1977 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*60*(1), 166-181. https://doi.org/10.1016/0022-247X(77)90057-9