A note on the existence of global solutions for reaction-diffusion   equations with almost-monotonic nonlinearities
  We show that reaction-diffusion equations with almost-monotonic nonlinear terms are well-posed in $L^q(\Omega)$ for each $1\leq q < \infty$ and the solutions are globally defined. 
An\'ibal Rodr\'iguez-Bernal and Alejandro Vidal-L\'opez
math.AP