2.4 混合问题

$$\begin{cases} u_{tt}-a^2u_{xx}=0, & Q=\{0<x<l,t>0\}\\ u(0,t)=u(l,t)=0, & t>0\\ u(x,0)=\varphi(x), & 0\leqslant x\leqslant l\\ u_t(x,0)=\psi(x), & 0\leqslant x\leqslant l. \end{cases}$$

称 $u(x,t)\in C(\overline{Q_T}) (Q_T=(0,l)\times (0,T))$ 为\hyperref[波动方程混合问题]{波动方程混合问题}的广义解, 当且仅当

对任意的 $\zeta(x,t)\in\mathscr D=\{\zeta(x,t)\in C^2(\overline{Q_T}):\zeta(x,T)=\zeta_t(x,T)=\zeta(0,t)=\zeta(l,t)=0\}$ 下述等式恒成立 $$ \iint_{Q_T}u(\zeta_{tt}-a^2\zeta_{xx})\text{d} x\text{d} t+\int_0^l\varphi\zeta_t(x,0)\text{d} x-\int_0^l\psi\zeta(x,0)\text{d} x=0. $$