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The thermoelasticity problem is solved for a gently sloping spherical shell acted on by a local heat source that moves over the shell surface. A linear distribution of the temperature over the shell depth and newtonian convective heat transfer from its surface are assumed. An analytic solution is obtained using integral Fourier and Laplace transforms. The distribution of the components of the stressed-deformed state of the shell is analyzed as a function of time, of the heat source velocity and shape, and of the heat transfer parameters.