The stress-strain state of a functionally gradient isotropic thin circular cylindrical shell under local heating by a flat heat source has been investigated. For this purpose, a mathematical model of the classical theory of inhomogeneous shells has been used. A two-dimensional heat equation is derived under the condition of a linear dependence of the temperature on the transverse coordinate. The solutions of the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a finite closed cylindrical pivotally supported shell have been obtained by means of methods of Fourier and Laplace integral transforms. Numerical results are presented for the metal-ceramic composite used to restore the integrity of human tooth crowns.