### Abstract

In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂_{t}u - Δu = Γ(u)∇u, ∇u) ⊥ T_{u}N, on M × [0, ∞) , u(t, x) ∈ Σ, for x ∈ ∂M, t > 0 , ∂u/∂n(t, x) ⊥ T_{u(t,x)} Σ, for x ∈ ∂M, t > 0 , u(o, x) = u_{o}(x), on M , where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝ^{n}, in which the nonlinear equation reduces to ∂_{t}u - Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than K_{N} ≤ 0 and Σ is totally geodesic in N.

Original language | English (US) |
---|---|

Pages (from-to) | 196-197 |

Number of pages | 2 |

Journal | Journal of Geometric Analysis |

Volume | 8 |

Issue number | 2 |

State | Published - 1998 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Geometric Analysis*,

*8*(2), 196-197.

**Evolution equations with a free boundary condition.** / Chen, Yunmei; Lin, Fang-Hua.

Research output: Contribution to journal › Article

*Journal of Geometric Analysis*, vol. 8, no. 2, pp. 196-197.

}

TY - JOUR

T1 - Evolution equations with a free boundary condition

AU - Chen, Yunmei

AU - Lin, Fang-Hua

PY - 1998

Y1 - 1998

N2 - In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂tu - Δu = Γ(u)∇u, ∇u) ⊥ TuN, on M × [0, ∞) , u(t, x) ∈ Σ, for x ∈ ∂M, t > 0 , ∂u/∂n(t, x) ⊥ Tu(t,x) Σ, for x ∈ ∂M, t > 0 , u(o, x) = uo(x), on M , where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝn, in which the nonlinear equation reduces to ∂tu - Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than KN ≤ 0 and Σ is totally geodesic in N.

AB - In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂tu - Δu = Γ(u)∇u, ∇u) ⊥ TuN, on M × [0, ∞) , u(t, x) ∈ Σ, for x ∈ ∂M, t > 0 , ∂u/∂n(t, x) ⊥ Tu(t,x) Σ, for x ∈ ∂M, t > 0 , u(o, x) = uo(x), on M , where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝn, in which the nonlinear equation reduces to ∂tu - Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than KN ≤ 0 and Σ is totally geodesic in N.

UR - http://www.scopus.com/inward/record.url?scp=29044449203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=29044449203&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:29044449203

VL - 8

SP - 196

EP - 197

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 2

ER -