The existence and stability of periodic solutions of inhomogeneous hyperbolic equation (Q2703862)

This paper concerns classical time-periodic solutions to the problem NEWLINE\[NEWLINEu_{tt}- u_{xx}+ bu_t+ cu= f(x,t),\quad x\in [0,\pi],NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0, t)= u(\pi, t)= 0,NEWLINE\]NEWLINE where \(b> 0\) and \(c\geq 0\) are constants and \(f(x,\cdot)\) is \(T\)-periodic. The following is proved: If \(f\) is continuous with respect to \(x\), \(C^2\)-smooth with respect to \(t\), and if NEWLINE\[NEWLINEf(t,x)= f_0(x)+ \sum^\infty_{n=1} f_n(x)\sin{2\pi_n\over T} t,NEWLINE\]NEWLINE then there exists exactly one (\(T\)-periodic with respect to time) solution, this solution is globally asymptotically stable, and NEWLINE\[NEWLINEu(t,x)= u_0(x)+ \sum^\infty_{n=1} \Biggl(v_n(x)\sin {2\pi_n\over T} t+ w_n(x)\cos{2\pi_n\over T} t\Biggr)NEWLINE\]NEWLINE with \(-u''_0+ cu_0= f_0\), \(v''_n+ [({2\pi_n\over T})^2- c] v_n+ b{2\pi_n\over T} w_n= f_n\), \(w''_n+ [({2\pi_n\over T})^2- c] w_n- b{2\pi_n\over T} v_n= 0\) and \(u_0(0)= u_0(\pi)= v_n(0)= v_n(\pi)= w_n(0)= w_n(\pi)= 0\).