A note on the paper by Qi S. Zhang: A priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem (Q1974237)

The author considers nonnegative solutions of the inhomogeneous heat equation \(u_t - \Delta u=f \geq 0\) on \(\mathbb{R}^n \times ] 0,T[\), with the initial condition that \(u(.,0)=u_0 \geq 0\) in the sense of local \(L^1\) convergence. Under certain smoothness conditions on \(f, u_0\) and \(u\), and assuming the existence of \(u\), he establishes the representation of \(u\) as the sum of the Gauss-Weierstrass integral of \(u_0\) and the heat potential of \(f\). Since \(u\) is a nonnegative supertemperature, the Riesz decomposition theorem plus the Widder representation theorem would almost have given the result. Among the closely related results which the author does not mention, perhaps the most general were obtained by the reviewer [J. Math. Anal. Appl. 67, 513-524 (1979; Zbl 0408.35045)]. For the paper mentioned in the title see ibid. 232, No. 2, 413-427, Art No. ID jmaa. 1999.6296 (1999; Zbl 0926.35058)].