Regularisation by multiplicative noise for reaction–diffusion equations

We consider the stochastic reaction–diffusion equation in $$1+1$$ 1 + 1 dimensions driven by multiplicative space–time white noise, with a distributional drift belonging to a Besov–Hölder space with any regularity index strictly larger than $$-1$$ - 1 . We assume that the diffusion coefficient is a regular function which is bounded away from zero. By using a combination of stochastic sewing techniques and Malliavin calculus, we show that the equation admits a unique solution.