On an evolution problem associated to the modelling of incertitude into the environment

We consider a mathematical model, posed by J.E. Scheinkman, simulating that an industrial project takes place into the environment without destroying it. We introduce a change of variable leading the formulation to a nonlinear evolution problem which we study by means of L∞L∞-accretive operator techniques. We prove that under suitable conditions there is extinction in finite time, which corresponds to some special behaviour of the solution of the original stochastic control problem.