Integral Recovery of a Time-Dependent Burgers Coefficient

We study an inverse coefficient problem for the one-dimensional periodic viscous Burgers equation, in which an unknown time-dependent coefficient multiplies the solution. Additional data are provided by a weighted spatial integral of the state. The periodic weight has first-order Sobolev regularity, and the prescribed observation is bounded away from zero. No sign restriction is imposed on the coefficient. An explicit reconstruction formula reduces the inverse problem to a loaded Burgers equation with a cubic nonlocal term. Bihari’s inequality controls the resulting superlinear energy estimate and yields a sufficient local existence threshold depending on the data. Using Fourier-Galerkin approximations, a priori estimates, and compactness arguments, we prove existence and uniqueness of a local strong solution and square-integrability in time of the reconstructed coefficient. The prescribed integral observation is recovered under the initial compatibility condition. A numerical experiment with a known exact solution and a sign-changing coefficient illustrates reconstruction accuracy and shows observed second-order spatial convergence for both the state and the coefficient.