Decay estimates for some semilinear damped hyperbolic problems
Decay estimates for some semilinear damped hyperbolic problems
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DOI：
10.1007/bf00282203
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发表时间：
1988-06
影响因子：
2.5
通讯作者：
A. Haraux;E. Zuazua
中科院分区：
文献类型：
--
作者：
A. Haraux;E. Zuazua
LetΩbe a bounded open domain inRn,g∶R→Ra non-decreasing continuous function such that g(0)=0 andhεLloc1(R+;L2(Ω)). Under suitable assumptions ongandh, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general formu′′+Lu+g(u′) =h(t,x)ast→ + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation $$u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega$$ inR+×Ω, u=0onR+×∂Ω. For instance ifwithc, d>0 and 1 <p≦q, (n−2)q≦n+2, then if, all solutions are bounded in the energy space for t≧0 and ifu, vare two such solutions, the energy norm ofu(t) − v(t)decays liket−1/p−1ast→ + ∞.
LetΩbe a bounded open domain inRn,g∶R→Ra non-decreasing continuous function such that g(0)=0 andhεLloc1(R+;L2(Ω)). Under suitable assumptions ongandh, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general formu′′+Lu+g(u′) =h(t,x)ast→ + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation $$u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega$$ inR+×Ω, u=0onR+×∂Ω. For instance ifwithc, d>0 and 1 <p≦q, (n−2)q≦n+2, then if, all solutions are bounded in the energy space for t≧0 and ifu, vare two such solutions, the energy norm ofu(t) − v(t)decays liket−1/p−1ast→ + ∞.