带第二声速的半线性热弹性力学方程组的间断解——变系数情况的Cauchy问题

Asymptotic Behavior of Discontinuous Solutions to the Semi-linear Thermoelastic Systems with Second Sound——Cauchy Problem to the Case of Variable Coefficients

对变系数的情况得到与常系数类似的结论:在松弛因子消失时,温度本身的间断将消失,而温度关于空间、时间的一阶偏导的间断会沿弹性波的特征传播;更进一步,在非线性项含有弹性波位移的一阶偏导时,如对其加一些特定增值条件,当t→∞时,以上提到的跳跃将按指数衰减,并且热传导系数越小衰减速度越快.具体算例的验算结果表明,当非线性函数不满足所给的增长条件时,解的间断跳跃度随时间发展将没有衰减性,从而说明此增长条件对保证解的间断跳跃度关于时间发展有指数衰减性是充分和必要的.

The Cauchy problem was considered for semilinear variable coefficient thermoelastic systems with second sound in one space dimension with discontinuous initial data. The constant coefficient case has been studied by Reinhard Racke and Wang Yaguang. By analyses, the conclusions similar were obtained that the jump of the temperature goes to zero while the jumps of the gradient of the displacement and the spatial derivative of the temperature are propagated along the hyperbolic characteristic curves when the relaxation parameter vanishes. Moreover, when f, g are independent of （ut,ux） or have certain growth restriction with respect to （ut,ux）, one deduces that these jumps decay exponentially when the time goes to infinity, more rapidly for small heat conduction coefficient. At the end, the paper introduced an example to show if the nonlinear functions f and g do not satisfy the growth restriction, the jumps of ut ,ux and θx will not decay exponentially when the time goes to infinity. So it indicates that the growth restriction is necessary and sufficient to guarantee the exponential decay of the jumps of solutions.