Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$
Fašangová, Eva
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998), p. 525-544 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 1998-01-01
Classification:  35B05,  35B40,  35K55 
@article{119030,
     author = {Eva Fa\v sangov\'a},
     title = {Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {39},
     year = {1998},
     pages = {525-544},
     zbl = {0963.35080},
     mrnumber = {1666798},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119030}
}

Fašangová, Eva. Asymptotic analysis for a nonlinear parabolic equation on $\Bbb R$. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 525-544. http://gdmltest.u-ga.fr/item/119030/

Feireisl E.; Petzeltová H. Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations 10 181-196 (1997). (1997) | MR 1424805

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). (1968) | MR 0223758

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

Chaffee N. A stability analysis for semilinear parabolic partial differential equation, J. Differential Equations 15 522-540 (1974). (1974) | MR 0358042

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). (1977) | MR 0442480 | Zbl 0361.35035

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

Protter M.H.; Weinberger H.F. Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1967. | MR 0219861 | Zbl 0549.35002

Arnold V.I. Ordinary Differential Equations, Nauka Moscow, 1971. | MR 0361231 | Zbl 0858.34001

Berestycki H.; Lions P.-L. Nonlinear Scalar Field Equations I, Existence of a Ground State, Arch. Rational Mech. Anal. 82 313-346 (1983). (1983) | MR 0695535 | Zbl 0533.35029

Reed M.; Simon B. Methods of Modern Mathematical Physics $4$, Academic Press New York-San Francisco-London, 1978. | MR 0493421

Coddington E.A.; Levinson N. Theory of Ordinar Differential Equations, McGraw-Hill New York-Toronto-London, 1955. | MR 0069338

Britton N.F. Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, 1986. | MR 0866143 | Zbl 0602.92001