2004/08/24 by Ross G. Pinsky, Ross Pinsky, Pinsky, Ross
Computer Science · Mathematics · Medicine · #35K15 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Partial Differential Equations #math.AP #msc:35K15 #msc:35K55
paper · pdf · doi:10.48550/arxiv.math/0408332
21 pages
arxiv created 2004/08/24 · openalex publication_date 2004/08/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider classical solutions to the parabolic reaction diffusion equation amp;ut =Lu+f(x,u), (x,t)∈ Rn×(0,∞); amp;u(x,0) =g(x)≥0, x∈ Rn; amp;u≥0, where L=∑i,j=1nai,j(x)(∂2)/(∂ xi ∂ xj)+∑i=1nbi(x)\frac∂∂ xi is a non-degenerate elliptic operator, g∈ C(Rn) and the reaction term f converges to -∞ at a super-linear rate as u→∞. 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) on Rn×(0,∞) such that u(x,t)≤ M(x,t), for all solutions to the Cauchy problem. Assuming now in addition that f(x,0)=0, so that u≡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=0 and the nonexistence of unbounded, stationary solutions to the corresponding elliptic problem. We also give a generic condition for nonexistence of nontrivial stationary solutions.