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We prove the existence and uniqueness of global solutions to the semilinearstochastic heat equation on an unbounded spatial domain with forcing terms thatgrow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$along with additional restrictions. For example, consider the forcing $f(u) = u\log(e^e + |u|)\log(\log(e^e+|u|))$. A new dynamic weighting procedure isintroduced to control the solutions, which are unbounded in space.

Existence and uniqueness of global solutions to the stochastic heat equation with super-linear drift on an unbounded spatial domain