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

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

2004-08-24

arxiv.org/abs/math/0408332v1

Mathematics

Analysis of PDEs

21 pages

Scientific paper

Consider 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.

Affiliated with

Also associated with

No associations

Rating

Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Reaction diffusion equations with super-linear absorption: universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-249556