### 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) |
---|---|

Pages (from-to) | 166-181 |

Number of pages | 16 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - 1977 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

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

**Existence and comparison theorems for nonlinear diffusion systems.** / Kuiper, Hendrik J.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 60, no. 1, pp. 166-181. https://doi.org/10.1016/0022-247X(77)90057-9

}

TY - JOUR

T1 - Existence and comparison theorems for nonlinear diffusion systems

AU - Kuiper, Hendrik J.

PY - 1977

Y1 - 1977

N2 - 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 (v1(x), v2(x),..., vm(x)) from Ω ⊂ Rn into Rm which satisfy ψi(x, t) ≤ vi(x) ≤ θi(x, t) for all x ε{lunate} Ω and 1 ≤ i ≤ m, where ψiand θi are extended real-valued functions on \ ̄gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),..., um(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.

AB - 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 (v1(x), v2(x),..., vm(x)) from Ω ⊂ Rn into Rm which satisfy ψi(x, t) ≤ vi(x) ≤ θi(x, t) for all x ε{lunate} Ω and 1 ≤ i ≤ m, where ψiand θi are extended real-valued functions on \ ̄gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),..., um(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.

UR - http://www.scopus.com/inward/record.url?scp=49449122533&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49449122533&partnerID=8YFLogxK

U2 - 10.1016/0022-247X(77)90057-9

DO - 10.1016/0022-247X(77)90057-9

M3 - Article

AN - SCOPUS:49449122533

VL - 60

SP - 166

EP - 181

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -