%0 Journal Article %1 MR705746 %A Bramson, Maury %D 1983 %J Mem. Amer. Math. Soc. %K Fisher-KPP reaction-diffusion travelling_wave %N 285 %P iv+190 %T Convergence of solutions of the Kolmogorov equation to travelling waves %U http://books.google.com/books?id=RLLFLt5skogC&dq=%22Convergence+of+solutions+of+the+Kolmogorov+equation+to+travelling+waves.+%22&printsec=frontcover&source=bl&ots=XYjJ1MszFN&sig=ZuR0LasI78mpykLBXafkLgJE0cA&hl=en&ei=TZCdSqTZLIH8tAP5xIkr&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false %V 44 %X The author studies the classical Kolmogorov equation $u_t=12u_xx+f(u)$ which arises in the context of a genetics model for the spread of an advantageous gene through a population. His main results are summarized in the following two theorems. Theorem A gives necessary and sufficient conditions on the initial data for convergence to the travelling waves $w^łambda(x)=u(t,x+t)$, $łambda\geq2$. Theorem A: Let $u(t,x)$ be a solution of $(*)$ $u_t=12u_xx+f(u)$ with $0u(0,x)1$, where $fC^10,1$ satisfies the conditions $f(0)=f(1)=0$, $f(u)>0$ for $0<u<1$, $f'(0)=1$, $f'(u)1$ for $0<u1$ and $1-f'(u)=O(u^\rho)$ for some $\rho>0$. Then (1) for $łambda>2$, $u(t,x+m(t))w^łambda(x)$ uniformly in $x$ as $t\rightarrowınfty$ for some choice of $m(t)$ if and only if for some (all) $h>0$ $$ łim_t\rightarrowınfty1tlogłeftınt_t^t(1+h)u(0,y)\,dy\right=-łambda+łambda^2-2\quadand\tag1\\ ınt_x^x+Nu(0,y)\,dy>ŋ\quadforxłeq-M\tag2 $$ for some $\eta>0$, $M>0$ and $N>0$. (ii) For the case $łambda=2$, $u(t,x+m(t))w^2(x)$ uniformly in $x$ as $t\rightarrowınfty$ for some choice of $m(t)$ if and only if for some (all) $h>0$ $$ łimsup_t\rightarrowınfty1tlogłeftınt_t^t(1+h)u(0,y)\,dy\rightłeq-2 $$ and condition (2) is satisfied. Theorem B gives a precise formula for $m(t)$ in the case $łambda>2$. Theorem B: Let $u(t,x)$ be a solution of $(*)$ as in Theorem A. If conditions (1) and (2) are satisfied, then $m(t)$ may be chosen so that $m(t)=\sup\x;\varphi(t,x)1\$ where $$ \varphi(t,x)=e^tınt_-ınfty^u(0,y)e^-(x-y)^2/2t2t\,dy. $$ The basic technique employed in the article is the use of the Feynman-Kac integral formula in conjunction with sample path estimates for Brownian motion. These estimates for the right tail of $u(t,x)$ are precise enough to allow comparison with the right tail of $w^łambda(x)$, $łambda\geq2$. Such comparison is sufficient to imply convergence to $w^łambda(x)$ as in Theorem A and enable us to compute the position of the wave as in Theorem B.

@article{MR705746, abstract = { The author studies the classical Kolmogorov equation $u_t=\frac 1{2}u_{xx}+f(u)$ which arises in the context of a genetics model for the spread of an advantageous gene through a population. His main results are summarized in the following two theorems. Theorem A gives necessary and sufficient conditions on the initial data for convergence to the travelling waves $w^\lambda(x)=u(t,x+\lambda t)$, $\lambda\geq\sqrt 2$. Theorem A: Let $u(t,x)$ be a solution of $(*)$ $u_t=\frac 1{2}u_{xx}+f(u)$ with $0\leq u(0,x)\leq 1$, where $f\in C^1[0,1]$ satisfies the conditions $f(0)=f(1)=0$, $f(u)>0$ for $0<u<1$, $f'(0)=1$, $f'(u)\leq 1$ for $0<u\leq 1$ and $1-f'(u)=O(u^\rho)$ for some $\rho>0$. Then (1) for $\lambda>\sqrt 2$, $u(t,x+m(t))\rightarrow w^\lambda(x)$ uniformly in $x$ as $t\rightarrow\infty$ for some choice of $m(t)$ if and only if for some (all) $h>0$ $$ \gather \lim_{t\rightarrow\infty}\frac 1{t}log\left[\int_t^{t(1+h)}u(0,y)\,dy\right]=-\lambda+\sqrt{\lambda^2-2}\quad\text{and}\tag1\\ \int_x^{x+N}u(0,y)\,dy>ŋ\quad\text{for}\quad x\leq-M\tag2 \endgather $$ for some $\eta>0$, $M>0$ and $N>0$. (ii) For the case $\lambda=\sqrt 2$, $u(t,x+m(t))\rightarrow w^{\sqrt 2}(x)$ uniformly in $x$ as $t\rightarrow\infty$ for some choice of $m(t)$ if and only if for some (all) $h>0$ $$ \limsup_{t\rightarrow\infty}\frac 1{t}log\left[\int_t^{t(1+h)}u(0,y)\,dy\right]\leq-\sqrt 2 $$ and condition (2) is satisfied. Theorem B gives a precise formula for $m(t)$ in the case $\lambda>\sqrt 2$. Theorem B: Let $u(t,x)$ be a solution of $(*)$ as in Theorem A. If conditions (1) and (2) are satisfied, then $m(t)$ may be chosen so that $m(t)=\sup\{x;\varphi(t,x)\geq 1\}$ where $$ \varphi(t,x)=e^t\int_{-\infty}^\infty u(0,y)\frac{e^{-(x-y)^2/2t}}{\sqrt{2\pi t}}\,dy. $$ The basic technique employed in the article is the use of the Feynman-Kac integral formula in conjunction with sample path estimates for Brownian motion. These estimates for the right tail of $u(t,x)$ are precise enough to allow comparison with the right tail of $w^\lambda(x)$, $\lambda\geq\sqrt 2$. Such comparison is sufficient to imply convergence to $w^\lambda(x)$ as in Theorem A and enable us to compute the position of the wave as in Theorem B. }, added-at = {2009-09-01T23:22:27.000+0200}, author = {Bramson, Maury}, biburl = {https://www.bibsonomy.org/bibtex/28e4c7ae7f3471663a4974a4e6d505a38/peter.ralph}, coden = {MAMCAU}, description = {MR: Publications results for "MR Number=(705746)"}, fjournal = {Memoirs of the American Mathematical Society}, interhash = {c98166891bc417481ed309ca3431db82}, intrahash = {8e4c7ae7f3471663a4974a4e6d505a38}, issn = {0065-9266}, journal = {Mem. Amer. Math. Soc.}, keywords = {Fisher-KPP reaction-diffusion travelling_wave}, mrclass = {60J65 (35K55 92A10)}, mrnumber = {MR705746 (84m:60098)}, mrreviewer = {Kazuaki Taira}, number = 285, pages = {iv+190}, timestamp = {2013-09-12T22:23:01.000+0200}, title = {Convergence of solutions of the {K}olmogorov equation to travelling waves}, url = {http://books.google.com/books?id=RLLFLt5skogC&dq=%22Convergence+of+solutions+of+the+Kolmogorov+equation+to+travelling+waves.+%22&printsec=frontcover&source=bl&ots=XYjJ1MszFN&sig=ZuR0LasI78mpykLBXafkLgJE0cA&hl=en&ei=TZCdSqTZLIH8tAP5xIkr&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false}, volume = 44, year = 1983 }