Bootstrap closure and nonlinear stability boundaries — model 1

Given: The inequality in the Example on every compact interval; $E_0\ge 0$; $C,ϵ,\delta >0$; and dimensionless time $t\ge 0$.

Gronwall's integral inequality with variable and constant coefficients: if continuous $u,a,b$ on $[0,T]$ satisfy $u,b\ge 0$ and $u(t)\le a(t)+{\int }_0^{t}b(s)u(s)ds$, then for nondecreasing $a$, $u(t)\le a(t)\exp ({\int }_0^{t}b)$.

Real powers for positive bases, with the zero-base positive-exponent convention: for a positive base $x$, $x^{\alpha }=\exp (\alpha \log x)$ for every real $\alpha$.

Continuity and derivatives of positive-base real powers: on $x>0$, $(x^{\alpha }{)}^{\mathrm{\prime }}=\alpha x^{\alpha -1}$.

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$: the derivative of a composition is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$: for differentiable $G$ on a compact interval with integrable derivative, ${\int }_a^b{G}^{\mathrm{\prime }}(s)ds=G(b)-G(a)$.

The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$: $\exp (x)>0$ for every real $x$.

The exponential function is strictly increasing: $\exp$ is strictly increasing on $\mathbb{R}$.

The real exponential function and the number $e$ by a power series: the power-series definition gives $\exp (0)=1$.

A quantitative continuation and bootstrap contract: fix an evolution problem and solution class, a continuous nonnegative dimensionless energy $E$ of a solution, constants $A,ϵ,\eta >0$ with $2A{ϵ}^2<\eta$, and finitely many margin functions $m_j:[0,T_{\max })\to \mathbb{R}$ with thresholds ${\mu }_j>0$. The separately supplied continuation criterion extends a finite-endpoint solution in the same class whenever $E\le \eta$ and every $m_j\ge {\mu }_j$ throughout its interval. A solution is maximal if no agreeing extension past its finite right endpoint exists in that class.

Open-closed bootstrap continuation: for a maximal solution satisfying [F9]'s continuation contract, an improvement $E\le 2A{ϵ}^2\Rightarrow E\le A{ϵ}^2$ on every finite $[0,T]$ with $T<T_{\max }$, from $E(0)<A{ϵ}^2$, closes the bound; if every listed margin persists throughout $[0,T_{\max })$, then $T_{\max }=\mathrm{\infty }$.