Your question is not very clear. However, judging by your comments I think you mean that the inner edge of the annulus is held at one temperature $T_1$ while the outer edge is held at a higher temperature $T_2$.
There will be a temperature gradient along the radius of the disk. We cannot assume that this temperature gradient will be linear. To find it we must solve the Laplace Equation for heat transfer with circular symmetry.
Unlike the case in which the annulus is heated uniformly, the temperature gradient will set up thermal stresses in the annulus. Each infinitessimally thin concentric ring will be pulled outwards and inwards by adjacent rings, the difference being balanced by elastic tension in that ring.
This is not an easy calculation, though not impossible. It could certainly be solved numerically.
Without any calculation we can say that the inner radius will increase, because it is pulled outwards by adjacent expanding material, although it will not increase as much as when the annulus is heated uniformly. The material expands most where the temperature greatest, which is nearest the outer rim. This means that the hole will be distorted out of its circular shape if it is not concentric with the outer rim.