TY - JOUR

T1 - Evolution equations with a free boundary condition

AU - Chen, Yunmei

AU - Lin, Fang Hua

N1 - Funding Information:
Acknowledgements and Notes. The research of the first author is partially supported by an NSF grant. The research of the second author is partially supported by an NSF-PYI grant.

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.

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U2 - 10.1007/bf02921640

DO - 10.1007/bf02921640

M3 - Article

AN - SCOPUS:29044449203

SN - 1050-6926

VL - 8

SP - 179

EP - 197

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 2

ER -