Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

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dc.creatorPinsky, Ross
dc.date2004-08-24
dc.date.accessioned2026-07-07T05:11:31Z
dc.date.available2026-07-07T05:11:31Z
dc.descriptionConsider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n; &u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i \partial x_j}+\sum_{i=1}^nb_i(x)\frac\partial{\partial x_i} $$ is a non-degenerate elliptic operator, $g\in C(R^n)$ and the reaction term $f$ converges to $-\infty$ at a super-linear rate as $u\to\infty$. 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 $R^n\times(0,\infty)$ such that $u(x,t)\le M(x,t)$, for all solutions to the Cauchy problem. Assuming now in addition that $f(x,0)=0$, so that $u\equiv0$ 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.
dc.description21 pages
dc.identifierhttps://arxiv.org/abs/math/0408332
dc.identifierhttp://arxiv.org/abs/math/0408332
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/72271
dc.subjectAnalysis of PDEs
dc.subject35K15;35K55
dc.titleReaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions
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