Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda_i\) satisfy
\(\Re\lambda_i<0\) for all \(i=1,\ldots,n\). Consider the initial value problem for
\(x_0\in\R^n\),
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=Ax+\dfrac{t\norm{x}^3}{1+t}v,\\
    x(0)=x_0
  \end{cases}
  \tag{2}
\end{equation}
where \(v\in\R^n\) is a unit vector.
\begin{enumerate}
\item[(a)] Show that if \(\lvert x_0\rvert\) is small enough, there is a bounded solution
to (2) that exists for all time.

[Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and \(c_0\ge0\) is a constant
that satisfy \(u(t)\le c_0+\int_0^t g(s)u(s)\dd s\) for all \(t\ge0\). Then Gronwall's
Inequality implies
\[
  u(t)\le c_0\exp\left(\int_0^t g(s)\dd s\right)
\]
for all \(t\ge0\).]
\item[(b)] Let \(z(t)=0\) for all \(t\) be the zero solution to (2) . Define what it means
for \(z\) to be \textit{Lyapunov Stable} (some authors call this notion \textit{stable} .)
Show that \(z\) is Lyapunov stable for this equation.
\end{enumerate}