Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Abstract: Consider classical solutions to the parabolic reaction diffusion equation whereis a nondegenerate elliptic operator, and the reaction term f converges to at a super-linear rate as . The first result in this paper is a parabolic Osserman–Keller type estimate. We give a sharp minimal growth condition on f, independent of L, in order that there exist a universal, a priori upper bound for all solutions to the above Cauchy problem—that is, in order that there exist a finite function on such that , for all solutions to the Cauchy problem. Assuming now in addition that , so that is a solution to the Cauchy problem, we show that under a similar growth condition, an intimate relationship exists between two seemingly disparate phenomena—namely, uniqueness for the Cauchy problem with initial data and the nonexistence of unbounded, stationary solutions to the corresponding elliptic problem. We also give a generic sufficient condition guaranteeing uniqueness for the Cauchy problem. [Copyright &y& Elsevier]