| Title:
|
Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$ (English) |
| Author:
|
Fašangová, Eva |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
39 |
| Issue:
|
3 |
| Year:
|
1998 |
| Pages:
|
525-544 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that nonnegative solutions of $$ \begin{aligned} & u_{t}-u_{xx}+f(u)=0,\quad x\in \Bbb R,\quad t>0, \\ & u=\alpha \bar u,\quad x\in \Bbb R,\quad t=0, \quad \operatorname{supp}\bar u \hbox{ compact } \end{aligned} $$ either converge to zero, blow up in $\operatorname{L}^{2}$-norm, or converge to the ground state when $t\to\infty$, where the latter case is a threshold phenomenon when $\alpha>0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $\alpha>0$, provided $\operatorname{supp}\bar u$ is sufficiently small. (English) |
| Keyword:
|
parabolic equation |
| Keyword:
|
stationary solution |
| Keyword:
|
convergence |
| MSC:
|
35B05 |
| MSC:
|
35B40 |
| MSC:
|
35K55 |
| idZBL:
|
Zbl 0963.35080 |
| idMR:
|
MR1666798 |
| . |
| Date available:
|
2009-01-08T18:45:59Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119030 |
| . |
| Reference:
|
[1] Feireisl E., Petzeltová H.: Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations.Differential Integral Equations 10 181-196 (1997). MR 1424805 |
| Reference:
|
[2] Zelenyak T.I.: Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable (in Russian).Differentsialnye Uravneniya 4 34-45 (1968). MR 0223758 |
| Reference:
|
[3] Feireisl E., Poláčik P.: Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on $\Bbb R$.Adv. Differential Equations, submitted, 1997. MR 1750112 |
| Reference:
|
[4] Chaffee N.: A stability analysis for semilinear parabolic partial differential equation.J. Differential Equations 15 522-540 (1974). MR 0358042 |
| Reference:
|
[5] Fife P.C., McLeod J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions.Arch. Rational Mech. Anal. 65 335-361 (1977). Zbl 0361.35035, MR 0442480 |
| Reference:
|
[6] Henry D.: Geometric Theory of Semilinear Parabolic Equations.Lecture Notes in Mathematics 840 Springer-Verlag Berlin-Heidelberg-New York, 1981. Zbl 0663.35001, MR 0610244 |
| Reference:
|
[7] Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations.Prentice Hall, Englewood Cliffs, N.J., 1967. Zbl 0549.35002, MR 0219861 |
| Reference:
|
[8] Arnold V.I.: Ordinary Differential Equations.Nauka Moscow, 1971. Zbl 0858.34001, MR 0361231 |
| Reference:
|
[9] Berestycki H., Lions P.-L.: Nonlinear Scalar Field Equations I, Existence of a Ground State.Arch. Rational Mech. Anal. 82 313-346 (1983). Zbl 0533.35029, MR 0695535 |
| Reference:
|
[10] Reed M., Simon B.: Methods of Modern Mathematical Physics $4$.Academic Press New York-San Francisco-London, 1978. MR 0493421 |
| Reference:
|
[11] Coddington E.A., Levinson N.: Theory of Ordinar Differential Equations.McGraw-Hill New York-Toronto-London, 1955. MR 0069338 |
| Reference:
|
[12] Britton N.F.: Reaction-Diffusion Equations and Their Applications to Biology.Academic Press, 1986. Zbl 0602.92001, MR 0866143 |
| . |
