A priori estimates of generalized solutions of the wave equation (Q1603137)

This paper is devoted to obtain a priori estimates for generalized solutions of the wave equation \[ u_{tt}(x,t)- u_{xx}(x,t)= 0\tag{1} \] in the spaces \(W^1_2\) and \(L_2\) with zero initial conditions \(u(x,0)=0\) and \(u_t(x,0)=0\) for \(0\leq x\leq \ell\). The author considers two cases: a) both boundary conditions \(u(0,t)= \mu(t)\), \(u(\ell,t)= \nu(t)\) are nonzero for \(0\leq t\leq T\), and \(T\leq\ell\); b) the right endpoint is fixed \(u(\ell,t)=0\), and the boundary condition \(u(0,t)=\mu(t)\) at the left endpoints is nonzero.