Positivity, Trotter products, and blow-up (Q702031)

In this work, the authors study blow up results for the nonlinear evolution equation \[ u^{\prime }(t)=Au(t)+\Phi (u(t)),\tag{1} \] where \(A\) is an \(m\)-dissipative generator of a positive semigroup on an ordered Banach space \(X\) and the function \(\Phi \) is nonlinear. Using the Lie--Trotter product formula, the authors establish results on the lower and upper bounds of solutions. Several examples are given to show how positivity can be used in order to prove the blow up results.