Advances in Differential Equations

Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on ${\bf R}$

Eduard Feireisl and Peter Poláčik

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider nonautonomous reaction-diffusion equations on $\mathbb R$, $$u_t-u_{xx}=f(t,u),\ x\in \mathbb R,\ t>0,$$ and investigate solutions in ${C_0(\mathbb R )}$, that is, solutions that decay to zero as $|x|$ approaches infinity. The nonlinearity is assumed to be a $C^1$ function which is $\tau$-periodic in $t$ and satisfies $\ f(t,0)=0\ $ and $ f_u(t,0) <0$ \ \ ($t\in \mathbb R$). Our two main results describe the structure of $\tau$-periodic solutions and asymptotic behavior of general solution with compact trajectories: (1) Each nonzero $\tau$-periodic solution is of definite sign and is even in $x$ about its unique peak (which is independent of $t$) . Moreover, up to shift in space, ${C_0(\mathbb R )}$ contains at most one $\tau$-periodic solution of a given sign. (2) Each solution with trajectory relatively compact in ${C_0(\mathbb R )}$ converges to a single $\tau$-periodic solution. In the proofs of these properties we make extensive use of nodal and symmetry properties of solutions. In particular, these properties are crucial ingredients in our study of the linearization at a periodic solution and in our description of center and stable manifolds of periodic solutions. The paper also contains a section discussing various sufficient conditions for existence of nontrivial periodic solutions.

Article information

Source
Adv. Differential Equations Volume 5, Number 4-6 (2000), 583-622.

Dates
First available in Project Euclid: 27 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651341

Mathematical Reviews number (MathSciNet)
MR1750112

Zentralblatt MATH identifier
0987.35079

Subjects
Primary: 35K57: Reaction-diffusion equations
Secondary: 35B10: Periodic solutions 35D05 37L99: None of the above, but in this section

Citation

Feireisl, Eduard; Poláčik, Peter. Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on ${\bf R}$. Adv. Differential Equations 5 (2000), no. 4-6, 583--622. https://projecteuclid.org/euclid.ade/1356651341.


Export citation