%0 Journal Article %1 fife1977approach %A Fife, Paul C. %A McLeod, J. B. %D 1977 %J Archive for Rational Mechanics and Analysis %K wave_of_advance wave_speed %N 4 %P 335--361 %R 10.1007/BF00250432 %T The approach of solutions of nonlinear diffusion equations to travelling front solutions %U https://doi.org/10.1007/BF00250432 %V 65 %X The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut---uxx---∞;(u)=O, x∈(---∞, ∞) , in the case ∞(0)=∞(1)=0, ∞´x(0)<0, ∞´x(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.1.Let u(x, 0)=u0(x) satisfy 0≦u0≦1. Let \$\$a\backslash\_ = \backslashmathop \\backslashlim \backslashsup u0\\backslashlimits\_\x \backslashto - \backslashinfty \ \\backslashtext\(\\x\\backslashtext\), \\\backslashmathop \\backslashlim \backslashinf u0\\backslashlimits\_\x \backslashto \backslashinfty \ \\backslashtext\(\\x\\backslashtext\)\\\$\$. Then u approaches a translate of U uniformly in x and exponentially in time, if a− is not too far from 0, and a+ not too far from 1.2.Suppose \$\$\backslashint\backslashlimits\_\\backslashtext\0\\^\\backslashtext\1\\ \f\\backslashtext\(\\u\\backslashtext\)\\du\ > \\backslashtext\0\\\$\$. If a− and a+ are not too far from 0, but u0 exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts.3.Under certain circumstances, u approaches a ``stacked'' combination of wave fronts, with differing ranges.

@article{fife1977approach, abstract = {The paper is concerned with the asymptotic behavior as t {\textrightarrow} ∞ of solutions u(x, t) of the equation ut---uxx---∞;(u)=O, x∈(---∞, ∞) , in the case ∞(0)=∞(1)=0, ∞{\textasciiacutex}(0)<0, ∞{\textasciiacutex}(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.1.Let u(x, 0)=u0(x) satisfy 0≦u0≦1. Let {\$}{\$}a{\backslash}{\_} = {\backslash}mathop {\{}{\backslash}lim {\backslash}sup u0{\}}{\backslash}limits{\_}{\{}x {\backslash}to - {\backslash}infty {\}} {\{}{\backslash}text{\{}({\}}{\}}x{\{}{\backslash}text{\{}), {\}}{\}}{\backslash}mathop {\{}{\backslash}lim {\backslash}inf u0{\}}{\backslash}limits{\_}{\{}x {\backslash}to {\backslash}infty {\}} {\{}{\backslash}text{\{}({\}}{\}}x{\{}{\backslash}text{\{}){\}}{\}}{\$}{\$}. Then u approaches a translate of U uniformly in x and exponentially in time, if a− is not too far from 0, and a+ not too far from 1.2.Suppose {\$}{\$}{\backslash}int{\backslash}limits{\_}{\{}{\backslash}text{\{}0{\}}{\}}^{\{}{\backslash}text{\{}1{\}}{\}} {\{}f{\{}{\backslash}text{\{}({\}}{\}}u{\{}{\backslash}text{\{}){\}}{\}}du{\}} > {\{}{\backslash}text{\{}0{\}}{\}}{\$}{\$}. If a− and a+ are not too far from 0, but u0 exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts.3.Under certain circumstances, u approaches a ``stacked'' combination of wave fronts, with differing ranges.}, added-at = {2023-03-20T18:10:52.000+0100}, author = {Fife, Paul C. and McLeod, J. B.}, biburl = {https://www.bibsonomy.org/bibtex/2bfda4645160f92a34575cb4aa75ec785/peter.ralph}, day = 01, doi = {10.1007/BF00250432}, interhash = {0b2b18fdea00af6e99b6b64217732741}, intrahash = {bfda4645160f92a34575cb4aa75ec785}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, keywords = {wave_of_advance wave_speed}, month = dec, number = 4, pages = {335--361}, timestamp = {2023-03-20T18:10:52.000+0100}, title = {The approach of solutions of nonlinear diffusion equations to travelling front solutions}, url = {https://doi.org/10.1007/BF00250432}, volume = 65, year = 1977 }