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

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.