一类强阻尼非线性双曲型方程组解的存在性与整体衰减估计

Existence and decay estimate of global solution for a nonlinear hyperbolic system with strong damping

研究一类强阻尼非线性双曲型方程组中阻尼项Δuit的耗散作用.利用逐次逼近方法和一系列经典的估计, 证明了弱解的局部存在性和惟一性.然后利用能量方法, 借助于一个不等式supt≤s≤t+1(s)≤β((t)-(t+1) )给出了解的衰减估计.分析结果表明,阻尼项强烈地改变双曲型方程解的渐近行为,强阻尼双曲型方程组的能量E(t)随时间按指数衰减, 即E(t)≤Cexp(-λt).

The dissipative function of term Δuit in a class of nonlinear hyperbolic system with strong damping is studied. Firstly, by use of the successive approximation method and a series of classical estimates, the local existence and uniqueness of weak solution are proved. Then, using energy method and by means of an inequality sup(t &le s &le t+1) φ(s) &le β(φ(t)-φ(t + 1)), decay estimate of solution is put forward. The result indicates that the damping term drastically changes the asymptotic behavior of the hyperbolic equation. In other words, the energy E(t) of the nonlinear hyperbolic system decays exponentially in time, i.e., E(t) &le Cexp(-λt).