From spectral to nonlinear instability, part 2

Here I want to extend the discussion of nonlinear instability in the previous post to the infinite-dimensional case. To formulate the problem fix a Banach space $X$ . If the starting point is a system of PDE then finding the right $X$ might be a non-trivial task and the best choice might depend on which application we have in mind. We consider an abstract equation of the form $\frac{du}{dt}=Au+F(u)$ . Here $t\mapsto u(t)$ is a mapping from an interval to $X$ . $A$ is a linear operator which may be unbounded. In fact the case I am most interested in here is that where $A$ is the generator of a strongly continuous semigroup of linear operators on $X$ . Hence $A$ may not be globally defined on $X$ . It will be assumed that $F$ is globally defined on $X$ . This is not a restriction in the applications I have in mind although it might be in other interesting cases. In this setting it is necessary to think about how a solution should be defined. In fact we will define a solution of the above equation with initial value $v$ to be a continuous solution of the integral equation $u(t)=e^{tA}v+\int_0^t e^{(t-\tau)A}F(u(\tau))d\tau$ . Here $e^{tA}$ is not to be thought of literally as an exponential but as an element of the semigroup generated by $A$ . When $A$ is bounded then $e^{tA}$ is really an exponential and the solution concept used here is equivalent to usual concept of solution for an ODE in a Banach space. Nonlinear stability can be defined just as in the finite-dimensional case, using the norm topology. In general the fact that a solution remains in a fixed ball as long as it exists may not ensure global existence, in contrast to the case of finite dimension. We are attempting to prove instability, i.e. to prove that solutions leave a certain ball. Hence it is convenient to introduce the convention that a solution which ceases to exist after finite time is deemed to have left the ball. In other words, when proving nonlinear instability we may assume that the solution $u$ being considered exists globally in the future. What we want to show is that there is an $\epsilon>0$ such that for any $\delta>0$ there are solutions which start in the ball of radius $\delta$ about $x_0$ and leave the ball of radius $\epsilon$ about that point.

The discussion which follows is based on a paper called ‘Spectral Condition for Instability’ by Jalal Shatah and Walter Strauss. At first sight their proof looks very different from the proof I presented in the last post and here I want to compare them, with particular attention to what happens in the finite-dimensional case. We want to show that the origin is nonlinearly unstable under the assumption that the spectrum of $A$ intersects the half of the complex plane where the real part is positive. In the finite-dimensional case this means that $A$ has an eigenvalue with positive real part. The spectral mapping theorem relates this to the situation where the spectrum of $e^A$ intersects the exterior of the unit disk. In the finite-dimensional case it means that there is an eigenvalue with modulus $\mu$ greater than one. We now consider the hypotheses of the Theorem of Shatah and Strauss. The first is that the linear operator $A$ occurring in the equation generates a strongly continuous semigroup. The second is that the spectrum of $e^A$ meets the exterior of the unit disk. The third is that in a neighbourhood of the origin the nonlinear term can be estimated by a power of the norm greater than one. The proof of nonlinear instability is based on three lemmas. We take $e^\lambda$ to be a point of the spectrum of $e^A$ whose modulus is equal to the spectral radius. In the finite-dimensional case this would be an eigenvalue and we would consider the corresponding eigenvector $v$ . In general we need a suitable approximate analogue of this. Lemma 1 provides for this by showing that $e^\lambda$ belongs to the approximate point spectrum of $e^A$ . Lemma 2 then shows that there is a $v$ whose norm grows at a rate no larger than $e^{{\rm Re}\lambda t}$ and is such that the norm of the difference between $e^Av$ and $e^{{\rm Re}\lambda t}v$ can be controlled. Lemma 3 shows that the growth rate of the norm of $e^A$ is close to $e^{{\rm Re}\lambda t}$ . In the finite-dimensional case the proofs of Lemma 1 and Lemma 2 are trivial. In Lemma 3 the lower bound is trivial in the finite-dimensional case. To get the upper bound in that case a change of basis can be used to make the off-diagonal entries in the Jordan normal form as small as desired. This argument is similar to that used to treat $A_c$ in the previous post. In the theorem we choose a ball $B$ such that instability corresponds to leaving it. As long as a solution remains in that ball the nonlinearity is under good control. The idea is to show that as long as the norm of the initial condition is sufficiently small the contribution of the nonlinear term on the right hand side of the integral equation will remain small compared to that coming from the linearized equation, which is growing at a known exponential rate. The details are complicated and are of course the essence of the proof. I will not try to explain them here.

This entry was posted on May 21, 2021 at 9:03 am and is filed under analysis, dynamical systems, partial differential equations. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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