Periodic solutions to one-dimensional wave equation with piece-wise constant coefficients. (English) Zbl 0896.35075

Barbu, V.; Pavel, N. H.

J. Differ. Equations 132, No. 2, 319-337 (1996).

The authors consider the one-dimensional wave equation \[ u(x) y_{tt}(x,t)- (u(x) y_x(x, t))_x= f(x,t),\quad x\in (0,1),\quad t\in\mathbb{R} \] with piecewise constant coefficients \(u\), boundary conditions of mixed type \(-u(0) y_x(0,t)= g(t)\), \(y(1,t)= 0\) for \(t\in\mathbb{R}\) and with periodicity conditions \(y(x,t)= y(x,t+ T)\), \(y_t(x,t)= y_t(x,t+ T)\). For rational \(T\), the dependence of weak solutions on the data is obtained. The case, where \(T\) is irrational, is left as an open problem. Moreover, the following optimization problem is studied: to given \(y_0\) find a control function \(u\) which minimizes \(\int^T_0 (y(0,t)- y_0(t))^2dt\), where \(y\) is the weak solution for the problem above, with coefficients given by \(u\) and with \(f= 0\). In the last section, the authors study the semilinear version of the problem \[ u(x) y_{tt}(x,t)- (u(x) y_x(x,t))_x+ F(x, t,y(x,t))= 0,\quad x\in(0,1),\quad t\in\mathbb{R} \] with the nonlinear function \(F\) which satisfies linear growth conditions with respect to the variable \(y\).

Reviewer: H.-D.Alber (Darmstadt)

MSC:

35L20	Initial-boundary value problems for second-order hyperbolic equations
35B10	Periodic solutions to PDEs
35R05	PDEs with low regular coefficients and/or low regular data
35L70	Second-order nonlinear hyperbolic equations

Keywords:

dependence of weak solutions on the data; optimization problem