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Here we design boundary feedback stabilizers to unbounded trajectories, forsemi-linear stochastic heat equation with cubic non-linearity. The feedbackcontroller is linear, given in a simple explicit form and involves only theeigenfunctions of the Laplace operator. It is supported in a given open subsetof the boundary of the domain. Via a rescaling argument, we transform thestochastic equation into a random deterministic one. Then, the simple form ofthe feedback, we propose here, allows to write the solution, of the randomequation, in a mild formulation via a kernel. Appealing to a fixed pointargument the existence \& stabilization result is proved.

Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation, driven by a multiplicative noise