For the stochastic heat equation on the torus with coefficients growing like u|log u|^A near zero (A<1 for drift, A<1/4 for noise), a unique global strictly positive mild solution exists.
Differential Equations 313 (2022)
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A stochastic heat equation with non-locally Lipschitz coefficients
For the stochastic heat equation on the torus with coefficients growing like u|log u|^A near zero (A<1 for drift, A<1/4 for noise), a unique global strictly positive mild solution exists.