[Paper Review] On the blow-up for critical semilinear wave equations with damping in the scattering case

Kyouhei Wakasa, Borislav Yordanov|arXiv (Cornell University)|Jul 17, 2018

Advanced Mathematical Physics Problems10 references3 citations

TL;DR

This paper establishes the blow-up of solutions to the critical semilinear wave equation with time-dependent scattering damping in $ℝ^n$ for $n \geq 2$, proving that solutions with small initial data blow up in finite time when the power $p$ equals the Strauss exponent $p_0(n)$. The proof extends prior sub-critical results using energy estimates and asymptotic analysis of fundamental solutions to the associated linearized equation with damping.

ABSTRACT

We consider the Cauchy problem for semilinear wave equations with variable coefficients and time-dependent scattering damping in $\mathbf{R}^n$, where $n\geq 2$. It is expected that the critical exponent will be Strauss' number $p_0(n)$, which is also the one for semilinear wave equations without damping terms. Lai and Takamura (2018) have obtained the blow-up part, together with the upper bound of lifespan, in the sub-critical case $p

Motivation & Objective

• To resolve the open problem of finite-time blow-up for semilinear wave equations with scattering damping at the critical exponent $p = p_0(n)$.
• To extend the lifespan upper bound estimates from the sub-critical case ($p < p_0(n)$) to the critical case ($p = p_0(n)$).
• To analyze the behavior of fundamental solutions to the damped wave equation with variable coefficients and time-dependent damping $a(t)$ satisfying $\int_0^\infty a(t)dt < \infty$.
• To establish two-sided bounds on the fundamental solutions $y_1(t,s;\lambda)$ and $y_2(t,s;\lambda)$ of the linearized equation $L_a y = \lambda^2 y$.
• To verify that the critical exponent for blow-up remains the Strauss exponent $p_0(n)$ even under scattering damping, confirming the conjecture in the critical case.

Proposed method

• Derives two-sided bounds for the fundamental solution $y_1(t,s;\lambda)$ of the linearized damped wave equation using the identity $(y_1' e^{A(t)})' = \lambda^2 y_1 e^{A(t)}$, where $A(t) = \int_0^t a(r)dr$.
• Establishes a lower bound for $y_1(t,s;\lambda)$ via the identity $\left(y_1 e^{A(t)} - \int_s^t a(r)y_1 e^{A(r)} dr\right)'' = \lambda^2 \left(y_1 e^{A(t)} - \int_s^t a(r)y_1 e^{A(r)} dr\right)$.
• Uses the Wronskian identity $y_2 y_1' - y_2' y_1 = e^{A(s)-A(t)}$ to derive bounds on $y_2(t,s;\lambda)$ and its ratio to $y_1(t,s;\lambda)$.
• Applies asymptotic comparison with hyperbolic functions: $\cosh\lambda(t-s)$ and $\sinh\lambda(t-s)/\lambda$, under the condition $\|a\|_{L^1} < \infty$.
• Derives the key inequality $e^{-\|a\|_{L^1}} \cosh\lambda(t-s) \leq y_1(t,s;\lambda) \leq \cosh\lambda(t-s)$, showing $y_1$ behaves like $\cosh\lambda(t-s)$ up to a constant factor.
• Uses the derivative identities for $y_2(t,s;\lambda)$ to prove $e^{\|a\|_{L^1}} \frac{\sinh\lambda(t-s)}{\lambda} \geq y_2(t,s;\lambda) \geq e^{-2\|a\|_{L^1}} \frac{\sinh\lambda(t-s)}{\lambda}$, confirming the asymptotic behavior of $y_2$.

Experimental results

Research questions

• RQ1 Does the critical exponent for blow-up of semilinear wave equations with scattering damping remain the Strauss exponent $p_0(n)$?
• RQ2 Can the lifespan upper bound estimate for the sub-critical case ($p < p_0(n)$) be extended to the critical case ($p = p_0(n)$)?
• RQ3 How do the fundamental solutions of the damped wave equation behave asymptotically when the damping coefficient $a(t)$ is integrable over $[0,\infty)$?
• RQ4 What are the precise two-sided bounds for the solutions $y_1(t,s;\lambda)$ and $y_2(t,s;\lambda)$ of the linearized equation $L_a y = \lambda^2 y$?
• RQ5 Is the blow-up result for $p = p_0(n)$ sharp in the sense of matching the lifespan estimate from the sub-critical case?

Key findings

• The solution to the Cauchy problem for the damped wave equation with $p = p_0(n)$ blows up in finite time for small initial data, confirming the critical case of Strauss' conjecture.
• The lifespan $T_\varepsilon$ satisfies the upper bound $T_\varepsilon \leq C \varepsilon^{-2p(p-1)/\gamma(p,n)}$, matching the sub-critical estimate and suggesting sharpness.
• The fundamental solution $y_1(t,s;\lambda)$ satisfies $e^{-\|a\|_{L^1}} \cosh\lambda(t-s) \leq y_1(t,s;\lambda) \leq \cosh\lambda(t-s)$, showing it behaves like $\cosh\lambda(t-s)$ up to a multiplicative constant.
• The solution $y_2(t,s;\lambda)$ satisfies $e^{-2\|a\|_{L^1}} \frac{\sinh\lambda(t-s)}{\lambda} \leq y_2(t,s;\lambda) \leq e^{\|a\|_{L^1}} \frac{\sinh\lambda(t-s)}{\lambda}$, confirming its asymptotic behavior as $t-s \to \infty$.
• The Wronskian identity $y_2 y_1' - y_2' y_1 = e^{A(s)-A(t)}$ is instrumental in deriving the bounds on $y_2/y_1$ and thus on $y_2$ itself.
• The analysis confirms that the critical exponent for blow-up remains $p_0(n)$ under scattering damping, extending the result from the undamped case to the damped setting.