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The blow-up rate for a loglog non-scaling invariant semilinear wave equation

Tristan Roy, Hatem Zaag

TL;DR

This work analyzes finite-time blow-up for a semilinear wave equation with a loglog perturbation $f(u)=|u|^{p-1}u\,g(u)$ in the subconformal regime, where the equation is not scale-invariant. The authors introduce similarity variables around non-characteristic blow-up points and construct a Lyapunov framework, including a corrective term, to obtain robust energy bounds and time-averaged control of the rescaled solution $w$. They prove that the local blow-up rate is sandwiched by constants times the blow-up profile of the associated ODE $v''=|v|^{p-1}v\,g(v)$, and, in particular, that the upper and lower bounds are proportional to the ODE blow-up rate $v(t)\sim (T-t)^{-\frac{2}{p-1}}\log^{- rac{a}{p-1}}(-\log(T-t))$ as $t\to T^-$. A key novelty is handling the lack of scaling invariance via a carefully designed Lyapunov functional and detailed energy-interpolation estimates, extending sharp blow-up-rate results to non-scale-invariant nonlinearities (including the loglog perturbation).

Abstract

We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant.

The blow-up rate for a loglog non-scaling invariant semilinear wave equation

TL;DR

This work analyzes finite-time blow-up for a semilinear wave equation with a loglog perturbation in the subconformal regime, where the equation is not scale-invariant. The authors introduce similarity variables around non-characteristic blow-up points and construct a Lyapunov framework, including a corrective term, to obtain robust energy bounds and time-averaged control of the rescaled solution . They prove that the local blow-up rate is sandwiched by constants times the blow-up profile of the associated ODE , and, in particular, that the upper and lower bounds are proportional to the ODE blow-up rate as . A key novelty is handling the lack of scaling invariance via a carefully designed Lyapunov functional and detailed energy-interpolation estimates, extending sharp blow-up-rate results to non-scale-invariant nonlinearities (including the loglog perturbation).

Abstract

We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant.
Paper Structure (19 sections, 15 theorems, 153 equations)

This paper contains 19 sections, 15 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof of the lower bound in (\ref{['Eqn:BlowUpRate']})
  4. Design of a Lyapunov functional in similarity variables
  5. Estimate of derivative of the energy
  6. Corrective term
  7. A new functional: decreasing and nonnegativity properties
  8. Proposition \ref{['Prop:BoundPolynws']}: the proof
  9. Some intermediate results
  10. Polynomial pointwise bound of $w(s)$
  11. Improved estimate of derivative of $\mathcal{E}(w)$
  12. Decreasing property of a Lyapunov functional
  13. Bounds
  14. Proof of Proposition \ref{['Prop:BoundFinalWs']}
  15. Proof of Lemma \ref{['Lem:BoundAvPot']}
  16. ...and 4 more sections

Key Result

Theorem 1

(The blow-up rate for a $\log\log$ perturbation of the power nonlineairty). Let $u$ be a solution of equation equ-pert2 defined on some set $\mathcal{D}$defD for some $1$-Lipschitz function $x\longmapsto T(x)$. Let $x_{0}$ be a non-characteristic point. Then there exists $t_{0}(x_{0}) \in [ 0, T(x_{ for some $0< k(N,p,a)\le K$, where $K$ depends only on an upper bound on $T(x_0)$, $1/T(x_0)$, the

Theorems & Definitions (31)

  • Theorem 1
  • Proposition 2: Bounds on time averages of $(w,\partial_sw)$ in $H^1\times L^2(B(O,1))$
  • Proposition 3: Bounds on $(w,\partial_sw)$ in $H^1\times L^2(B(O,1))$
  • Lemma 4: Upper bound on the derivative of $\mathcal{E}(w(s))$ \ref{['Eqn:Energyw']}
  • proof
  • Lemma 5: Upper bound on the derivative of the corrective term
  • proof
  • Lemma 6: A nonnegative Lyapunov functional for equation \ref{['Eqn:Wavew']}
  • proof
  • Lemma 7
  • ...and 21 more