The Initial Value Problem For Evolution Equation On Manifolds With Nonlinear Damping And Source Terms

In this paper,we research the initial value problem of wave equation on manifolds with nonlinear damping and source terms utt+Au+|ut|mut = |u|ρu,(x,t)∈ Γ×((0,+∞)u(x,0)= u0(x),ut(x,0)＝u1(x),x ∈Γwhere Γ is the compact and smooth boundary of a domain Ω of Rn,m,ρ>0,A is a elliptic operator defined on manifold.By way of Faedo-Galerkin method,we obtain the local existence and uniqueness of regular solution in L∞(0,T0;H1/2(r))under the condition of 0<m,ρ1≤1/n-2(n≥ 3),m,ρ>0(n = 2),u0 ∈D(A),u1∈H1/2(r).Besides,When 0<ρ≤<m,we derive the global existence.When 0<m<ρ,a blow-up of solution with negative initial energy E(0)<0 is proved.Using the Faedo-Galerkin method and potential well theory,we prove the global existence and uniqueness of regular solution in Lloc∞(0,∞;H1/2(Γ))under the condition of E(0)<d,0<m,ρ≤1/n-2(n ≥3),m,ρ>0(n = 2),u0 ∈W∩D(A)(W denotes potential well),u1∈H1/2(r).Furthermore,we study the decay estimation of energy adapting the ideas introduced by Patrick Martinez who used these to investigate the decay rate for dissipative systems.