The Blow-Up Rate for Strongly Perturbed Semilinear Wave Equations

Abstract

We consider in this paper a large class of perturbed semilinear wave equation with subconformal power nonlinearity. In particular, we allow log perturbations of the main source. We derive a Lyapunov functional in similarity variables and use it to derive the blow-up rate. Throughout this work, we use some techniques developped for the unperturbed case studied by Merle and Zaag (Int. Math. Res. Notices, 19(1):1127–1156, 2005) together with ideas introduced by Hamza and Zaag in (Nonlinearity, 25(9):2759–2773, 2012) for a class of perturbations.

Appendix: Blow-Up Dynamics for the Associated ODE

In this appendix, we consider the following ODE:

$$\begin{aligned} u^{\prime \prime }=|u|^{p-1}u+f(u), \end{aligned}$$

(4.1)

with either $f(u)\sim |u|^q,\,q<p$ as $u\rightarrow \infty$ or $f(u)\sim \frac{|u|^p}{(\log (2+u^2))^a}$ , $a>1$ as $u\rightarrow \infty$.

In this proposition, we give two terms in the solution’s expansion near blow-up

Proposition 3.1

Let $u$ a solution of (4.1) that blows-up in some finite time $T$, the blow-up profile of $u$ near $T$ is given by the following quantities:

• (i) If $f(u)\sim |u|^q,$ then we have

  $$\begin{aligned} u(t)- \frac{\kappa }{(T-t)^{\frac{2}{p-1}}}\sim \frac{A}{(T-t)^{\frac{2}{p-1}-\mu }}, \qquad \mathrm as \quad t\rightarrow T^-, \end{aligned}$$

  (4.2)

  where $\mu >0$ and $A\in {\mathbb {R}}$.

• (ii) If $f(u)\sim \frac{|u|^p}{(\log (2+u^2))^a},$ then we have

  $$\begin{aligned} u(t)- \frac{\kappa }{(T-t)^{\frac{2}{p-1}}}\sim \frac{\kappa (p-1)^{a-1}}{4^a(T-t)^{\frac{2}{p-1}}(-\log (T-t))^a},\qquad \mathrm as \quad t\rightarrow T^-. \end{aligned}$$

  (4.3)

Remark 4

If $f(u)\equiv 0,$ then we have

$$\begin{aligned} u(t)- \frac{\kappa }{(T-t)^{\frac{2}{p-1}}}\sim \frac{A}{(T-t)^{\frac{2}{p-1}-2}}, \qquad \mathrm as \quad t\rightarrow T^-. \end{aligned}$$

Proof

First it is clear that $u(t)\sim \frac{\kappa }{(T-t)^{\frac{2}{p-1}}},$ as $t\rightarrow T^-,$ with $\kappa =(\frac{2p+2}{(p-1)^2})^{\frac{1}{p-1}}$. By using the following change of variables:

$$\begin{aligned} s=-\log (T-t), \quad u(t)=\frac{1}{(T-t)^{\frac{2}{p-1}}}w(-\log (T-t)),\quad \forall \quad t\in [0,T). \end{aligned}$$

The function $w$ satisfies the following equation: $\forall \quad s\ge -\log (T)$

$$\begin{aligned} w''(s)+\frac{p+3}{p-1}w'(s)+\frac{2p+2}{(p-1)^{2}}w(s)=|w(s)|^{p-1}w(s)+e^{\frac{-2ps}{p-1}}f(e^{\frac{2s}{p-1}}w(s)). \end{aligned}$$

(4.4)

By standard arguments, we easily study the asymptotic behaviour of equation (4.4) as $s\rightarrow \infty$ and we get (4.2) and (4.3). Which ends the proof of Proposition 3.1. $\square$