Critical case linear autonomous functional differential equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:35:55Z http://mathoverflow.net/feeds/question/116056 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116056/critical-case-linear-autonomous-functional-differential-equation Critical case linear autonomous functional differential equation PaweÅ‚ Biernat 2012-12-11T08:35:12Z 2012-12-12T15:26:49Z <p>I am looking for asymptotic ($t\to\infty$) behavior of the general solution $g(t)$ to a following linear functional differential equation</p> <p>$$\text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t)$$</p> <p>with initial data $g(t)=g_0(t)$ on $-T \leq t\leq 0$, let's say $g_0\in\mathcal C^n$.</p> <p>The asymptotics of solutions to</p> <p>$$\text{(2)} \quad\quad\quad g'(t)-Cg'(t-T)=Ag(t)+Bg(t-T),$$</p> <p>including $C=1$, is governed by the characteristic function</p> <p>$$H(\lambda)=\lambda(1-Ce^{-\lambda T})-A-Be^{-\lambda T}$$</p> <p>via the following theorem</p> <blockquote> <p>If <code>$\alpha_0=\sup\{Re(\lambda)\,:\,H(\lambda)=0\}$</code> and $g(t)$ is a solution to (2), then, for any $\alpha>\alpha_0$, there is a constant $K=K(\alpha)$ such that $$\lvert g(t)\rvert\le Ke^{\alpha t}\sup_{-T\le s\le0}\lvert g_0(s)\rvert.$$</p> </blockquote> <p>Because of the context in which equation (1) arises I expect $g(t)$ to converge to $0$, so let's apply the theorem.</p> <p>The roots of characteristic function of (1) are solutions to equation</p> <p>$$e^{\lambda T}=\frac{\lambda}{\lambda+1}.$$</p> <p>After taking the module of both sides we have</p> <p>$$e^{Re(\lambda) T}=\big\lvert\frac{\lambda}{\lambda+1}\big\rvert&lt;1,$$</p> <p>so $Re(\lambda)&lt;0$. 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:</p> <p><em>How to establish rate of convergence (or divergence) of solutions to equation (1) and its (in)dependence on initial data?</em></p> <p>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.</p> <p>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.</p>