Abstract: We consider wave equation with inhomogeneity $\sigma(x)u^m_t+q(x) u^p$, $m>1$, $p>1$, and study a forward and an one-dimensional inverse problems for this equation. \medskip Let $T$ be a real positive number. \medskip \noindent {\bf A forward problem.} Determine the function $u(x,t)$ satisfying the relations \begin{gather} u_{tt}-u_{xx} -\sigma(x)u^m_t -q(x) u^p=0,\quad x>0, \quad t\in(0,T]; \tag{1}\\ u|_{t=0}=u_t|_{t=0}=0, \tag{2}\\ u|_{x=0}=f(t), \tag{3} \end{gather} where $\sigma(x)$ and $q(x)$ are continuous functions; $m>1$ and $p>1$ are real numbers; $f(t)$ is the twice continuously differentiable function and $f(0)=a>0$, $f'(0)=b>0$, $a$, $b$ are some constants. \medskip \noindent {\bf An inverse problem.} Let $f_k(t)$, $k=1,2$, be the given functions and $f_k(0)=a_k>0$, $f'_k(0)=b_k$, $k=1,2$, numbers $a_k$ and $b_k$ satisfy the condition $a_1^pb_2^m-a_2^pb_1^m\neq0$. The solution of the forward problem (1)--(3) for $f=f_k$, $k=1,2$ denote $u^k(x,t)$, $k=1,2$. Find the functions $\sigma(x)$ and $q(x)$ from the given information about solutions $u^k(x,t)$: \begin{equation} u^k_x|_{x=0}=h_k(t),\quad t\in[0,T], \quad k=1,2. \tag{4} \end{equation} Conditions for the unique solvability of the forward problem are found. For the inverse problem a local existence and uniqueness theorems are established. Both theorems for forward and inverse problems are new in the theory of inverse problems (see paper~\cite{Romanov}). The work is done within framework of state contract of Sobolev institute of~Mathematis (project FWNF-2022-0009). \begin{ltrte} \bibitem{Romanov} {\it Romanov V.G., Bugueva T.V.} An one-dimensional inverse problem for the wave equation // Euras. J.~Math. Comput. Appl. 2024. V.~12. No.~3. P.~135--162. \end{ltrte}

Cite: Romanov V.G. , Bugueva T.V.
One-dimensional inverse problem for wave equation
In compilation Международная научная конференция «Современные проблемы обратных задач», посвященная 100-летию со дня рождения академика Г.И. Марчука. ТЕЗИСЫ. Институт математики им. С.Л. Соболева СО РАН, Математический центр в Академгородке. Новосибирск, Академгородок, 2 – 4 октября 2025 года. 35 С.. 2025. – C.29--30.

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