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

Consider classical solutions u∈C2(Rn×(0,∞))∩C(Rn×[0,∞))u∈C2(Rn×(0,∞))∩C(Rn×[0,∞)) to the parabolic reaction diffusion equation ut=Lu+f(x,u),(x,t)∈Rn×(0,∞),u(x,0)=g(x)⩾0,x∈Rn,u⩾0,whereL=∑i,j=1nai,j(x)∂2∂xi∂xj+∑i=1nbi(x)∂∂xiis a nondegenerate elliptic operator, g∈C(Rn)g∈C(Rn) and the reaction term f   converges to -∞-∞ at a super-linear rate as u→∞u→∞. 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 M(x,t)M(x,t) on Rn×(0,∞)Rn×(0,∞) such that u(x,t)⩽M(x,t)u(x,t)⩽M(x,t), for all solutions to the Cauchy problem. Assuming now in addition that f(x,0)=0f(x,0)=0, so that u≡0u≡0 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 g=0g=0 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.