Model structure on co-Segal commutative dg-algebras in characteristic p>0

We study weak commutative algebras in a symmetric monoidal model category $\mathscr{M}$. We provide a model structure on these algebras for any symmetric monoidal model category that is combinatorial and left proper. Our motivation was to have a homotopy theory of weak commutative dg-algebras in characteristic $p>0$, since there is no such theory for strict commutative dg-algebras. For a general $\mathscr{M}$, we show that if the projective model structure on strict commutative algebras exists, then the inclusion from strict to weak algebras is a Quillen equivalence. The results of this paper can be generalized to symmetric co-Segal $\mathcal{P}$-algebras for any operad $\mathcal{P}$. And surprisingly, the axioms of a monoidal model category are not necessary to get the model structure on co-Segal commutative algebras