On the existence of a solution to the inverse source problem for the Hopf equation
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In this paper, we consider a one-dimensional inverse source problem for the Hopf
equation. The problem consists in determining the unknown solution and the time-dependent source
term from the given initial condition and an additional overdetermination condition at a fixed spatial
point. By reducing the original inverse problem to a loaded equation and applying the fixed-point
method, we establish the existence of a solution in the class of functions of finite smoothness. The
proof is based on a priori estimates, the Schauder fixed-point theorem for smooth data, and a limit
transition argument using weak-* compactness for the general case. As a result, sufficient conditions
for the solvability of the inverse source problem are obtained
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