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This paper investigates a class of PDEs with coefficients in negative Besovspaces and whose solutions have linear growth. We show existence and uniquenessof mild and weak solutions, which are equivalent in this setting, and severalcontinuity results. To this aim, we introduce ad-hoc Besov-H{\"o}lder typespaces that allow for linear growth, and investigate the action of the heatsemigroup on them. We conclude the paper by introducing a special subclass ofthese spaces which has the useful property to be separable.

A pde with drift of negative Besov index and linear growth solutions