A property of the heat equation which extends to the thermoelastic equations (Q1081326)

The author studies some properties of smooth solutions to the one- dimensional equations of linear thermoelasticity in the strip \(D=\{(x,t)\in (0,1)\times (0,\infty)\}\). These properties extend analogous ones valid for the simple heat equation and which are an easy consequence of the maximum principle. He considers the initial-boundary value problem \(\theta_{xx}=\theta_ t+au_{xt}\), \(u_{xx}=a\theta_ x+bu_{tt}\), \(\theta (x,0)=\theta_ 0(x)\), \(\theta (0,t)=f(t)\), \(\theta (l,t)=g(t)\), \(u(x,0)=u_ 0(x)\), \(u_ t(x,0)=u_ 1(x)\), \(u(0,t)=u(l,t)=0\). Here \(\theta\) and u represent (dimensionless) temperature and displacement, respectively, while \(\theta_ 0\), \(u_ 0\), \(u_ 1\), f and g are prescribed functions of their arguments. Moreover, a and b are suitable dimensionless parameters. Set \(w^{(i)}=d^{(i)}w/dt^{(i)}\). Then the properties proved by the author are the following: Let \(\theta (x,t),u(x,t)\in C^ 3(D)\) be a solution to the above problem with \(a^ 2<2\). Assume f and g to be of class \(C^{n+3}\) for some \(n\leq 2\) and \(f^{(1)}\) and \(g^{(1)}\) both positive for all large t or both negative for all large t. Suppose in addition that \(f^{(i)}=o(| f^{(1)}|)\), \(g^{(i)}=o(| g^{(1)}|)\) \((i=2,...,n)\) as \(t\to \infty\) and the derivatives \(f^{(i)}\), \(g^{(i)}\) are integrable and square integrable \((i=n+1,n+2,n+3)\) over (0,\(\infty)\). Then \(\max_{0\leq x\leq 1}\theta (x,t)=(f+g)+| f-g| +o(1)\), \(\min_{0\leq x\leq 1}\theta (x,t)=(f+g)-| f-g| +o(1)\) according as \(f^{(1)}\) and \(g^{(1)}\) are positive or negative for all large t. The above propertie hold, whatever the initial distributions \(\theta_ 0\), \(u_ 0\) and \(u_ 1\).