Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
parabolic equation; stationary solution; convergence
parabolic equation; stationary solution; convergence
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.
References:
[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
[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
[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
[4] Chaffee N.: A stability analysis for semilinear parabolic partial differential equation. J. Differential Equations 15 522-540 (1974). MR 0358042
[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). MR 0442480 | Zbl 0361.35035
[6] Henry D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840 Springer-Verlag Berlin-Heidelberg-New York, 1981. MR 0610244 | Zbl 0663.35001
[7] Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs, N.J., 1967. MR 0219861 | Zbl 0549.35002
[8] Arnold V.I.: Ordinary Differential Equations. Nauka Moscow, 1971. MR 0361231 | Zbl 0858.34001
[9] Berestycki H., Lions P.-L.: Nonlinear Scalar Field Equations I, Existence of a Ground State. Arch. Rational Mech. Anal. 82 313-346 (1983). MR 0695535 | Zbl 0533.35029
[10] Reed M., Simon B.: Methods of Modern Mathematical Physics $4$. Academic Press New York-San Francisco-London, 1978. MR 0493421
[11] Coddington E.A., Levinson N.: Theory of Ordinar Differential Equations. McGraw-Hill New York-Toronto-London, 1955. MR 0069338
[12] Britton N.F.: Reaction-Diffusion Equations and Their Applications to Biology. Academic Press, 1986. MR 0866143 | Zbl 0602.92001
Search
Partner of
