Vibrating infinite string under general observation conditions and minimally smooth force
András Szijártó, Jenõ Hegedűs · Electronic journal of qualitative theory of differential equations · 2016
Existence of the classical solution $u(x,t)\in C^2(\mathbb{R}^2)$ to the Problem \eqref{eq:1}, \eqref{eq:2} (shortly Problem $\mathcal{A}$): \[\tag{1}\label{eq:1} L u := u_{tt}(x,t)-a^2 u_{xx}(x,t)=f(x,t), \qquad (x,t)\in \mathbb{R}^2, \ a>0, \] under the observation conditions (the observed states) given at $t_1, \ t_2 \in \mathbb{R}$ with variable coefficients $A_1, \ B_1, \ A_2, \ B_2$ such that \[\begin{aligned} A_1(x) u|_{t=t_1} + B_1(x) u_t|_{t=t_1}&=g_1(x), & \quad x&\in \mathbb{R}, \\ A_2(x) u|_{t=t_2} + B_2(x) u_t|_{t=t_2}&=g_2(x), & \quad x&\in \mathbb{R}, \end{aligned}\tag{2}\label{eq:2} \] is proved. Here the coefficients $A_i, \ B_i, \ i=1,2$, and $g_1, \ g_2$ are given functions smooth enough, $f\in C(\mathbb{R}^2)$, the directional derivative $ \partial f/ \partial t$ exists and $\partial f/ \partial t \in C(\mathbb{R}^2)$.