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I am looking for asymptotic ($t\to\infty$) behavior of the general solution $g(t)$ to a following linear functional differential equation

$$ \text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t) $$

with initial data $g(t)=g_0(t)$ on $-T \leq t\leq 0$, let's say $g_0\in\mathcal C^n$.

The asymptotics of solutions to

$$ \text{(2)} \quad\quad\quad g'(t)-Cg'(t-T)=Ag(t)+Bg(t-T), $$

including $C=1$, is governed by the characteristic function

$$ H(\lambda)=\lambda(1-Ce^{-\lambda T})-A-Be^{-\lambda T} $$

via the following theorem

If $\alpha_0=\sup\{Re(\lambda)\,:\,H(\lambda)=0\}$ and $g(t)$ is a solution to (2), then, for any $\alpha>\alpha_0$, there is a constant $K=K(\alpha)$ such that $$ > \lvert g(t)\rvert\le Ke^{\alpha t}\sup_{-T\le s\le0}\lvert g_0(s)\rvert. > $$

Because of the context in which equation (1) arises I expect $g(t)$ to converge to $0$, so let's apply the theorem.

The roots of characteristic function of (1) are solutions to equation

$$ e^{\lambda T}=\frac{\lambda}{\lambda+1}. $$

After taking the module of both sides we have

$$ e^{Re(\lambda) T}=\big\lvert\frac{\lambda}{\lambda+1}\big\rvert<1, $$

so $Re(\lambda)<0$. On the other hand, for $\lvert\lambda\rvert\gg1$ real part of $\lambda$ converges to $0-$, so $\alpha_0=0$ and the last theorem is inconclusive in the matter of convergence of $g(t)$ to $0$ so the question is:

How to establish rate of convergence (or divergence) of solutions to equation (1) and its (in)dependence on initial data?

The conjecture is that $g(t)\sim t^{\gamma}e^{-\beta t}$, where $\gamma$ and $\beta\ge0$ might depend on the differentiability class of initial data, but that's a wild guess.

PS I already posted the question on math.stackexchange, only then I read about mathoverflow and found out that it would be more suitable, so sorry for posting it twice.

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That depends on what kind of results you want to get. You are still in good shape if you want just the growing terms but the decaying case may be quite a mess. If you tell us more details, somebody may give you some ideas :). –  fedja Dec 12 '12 at 2:51
    
The conjecture is certainly false because you can design a convergent series of the exponential solution in which a bunch of several first terms will prevail for arbitrarily long but then the next term will start dominating. The corresponding data will be even $C^\infty$. On the other hand, high smoothness seems to imply at least a power decay with high power. If that's what you want, I'll elaborate :). –  fedja Dec 14 '12 at 11:01
    
Do I understand correctly, that you want to use $e^{\lambda_k t}$ with $H(\lambda_k)=0$ to show that no $\beta>0$ is possible? Then what would happen for infinite linear combination of such solutions? My feeling is that the asymptotics of $g(t)$ will depend on asymptotics of coefficients $a_k$ of such linear combination, but what's the form of this dependence? The other question is what assumptions on $g_0$ do we need, apart from $C^\infty$, to ensure that such linear combination will be infinite? Compact support inside $[-T,0]$? Please feel free to elaborate and correct me if I'm wrong. –  Paweł Biernat Dec 15 '12 at 11:03
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