Energy currents and conditional stability estimates — Examples

Let $\cosh x:=(e^x+e^{-x})\mathrm{/}2$, $w(x):=1\mathrm{/}\cosh x$, and $V(x):=-2w(x{)}^2$. These are smooth real functions. Put $\tau (x):=(e^x-e^{-x})\mathrm{/}(e^x+e^{-x})$; direct differentiation gives $w^{\mathrm{\prime }}=-w\tau$, ${\tau }^{\mathrm{\prime }}=1-{\tau }^2=w^2$, and $w^2=1-{\tau }^2$. Therefore $w^{\mathrm{\prime }\mathrm{\prime }}=w(2{\tau }^2-1)=w-2w^3$. Inductively each derivative $w^{(k)}$ is $w$ times a polynomial $P_k(\tau )$; since $\mathrm{\mid }\tau \mathrm{\mid }\le 1$ and $\cosh x\ge \frac{1}{2}{e}^{\mathrm{\mid }x\mathrm{\mid }}$, there is a finite constant $C_k$ with $\mathrm{\mid }{w}^{(k)}(x)\mathrm{\mid }\le C_k{e}^{-\mathrm{\mid }x\mathrm{\mid }}$ for all $x$. In particular, the integrals ${\int }_{\mathbb{R}}\mathrm{\mid }w{\mathrm{\mid }}^{2}dx$ and ${\int }_{\mathbb{R}}(\mathrm{\mid }w{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2)dx$ are finite, so the stated $L^2$ and $H^1$ norms are finite. Consequently the differential expression $A:=-{\mathrm{\partial }}_x^2+V(x)$ satisfies $Aw=-w$.

Define $U(t,x)=e^{t}w(x)$. Then $U_{tt}=e^{t}w$ and $-U_{xx}+VU=e^t(-w^{\mathrm{\prime }\mathrm{\prime }}-2w^3)=-e^{t}w$ by step 1.1. For every compact $K\subset \mathbb{R}$, $e^t$ is bounded on $K$; for every integer $m\ge 0$, $(1+\mathrm{\mid }x\mathrm{\mid }{)}^m{e}^{-\mathrm{\mid }x\mathrm{\mid }}$ is bounded on $\mathbb{R}$. The derivative bounds from step 1.1 therefore give the exact local-uniform decay condition in the refuted Statement for every spatial derivative of $U$. Thus $U$ is an admissible solution of that assertion. Therefore $U_{tt}-U_{xx}+VU=0$, while $\mathrm{\parallel }U(t){\mathrm{\parallel }}_{H^1}+\mathrm{\parallel }{U}_t(t){\mathrm{\parallel }}_2=e^t(\mathrm{\parallel }w{\mathrm{\parallel }}_{H^1}+\mathrm{\parallel }w{\mathrm{\parallel }}_2)⟶\mathrm{\infty }\mathrm{.}$ The initial norm is finite and nonzero, so no time-uniform constant $C(V,N)$ as asserted exists. The example refutes only the claim that symmetry alone suffices; its potential violates the positive-semidefinite hypothesis of the supplied theorem and is not asserted to be an Einstein–Maxwell master potential.