Global solutions with asymptotic self-similar behaviour for the cubic wave equation

Abstract

We construct a two-parameter family of explicit solutions to the cubic wave equation on $\mathbb{R}^{1+3}$. Depending on the value of the parameters, these solutions either scatter to linear, blow-up in finite time, or exhibit a new type of threshold behaviour which we characterize precisely.

Figures (7)

• Figure 1: Illustration of Theorem \ref{['thm:main']}; blow-up, scattering and asymptotically self-similar behaviour for $t>0$.
• Figure 2: The nine possible behaviors in the plane $(X, Y)$. The dots mark the four unique points that give self-similar behavior at $\pm\infty$.
• Figure 3: If $\lvert T_\pm\rvert<\pi$, then $u=u(t, \lvert x \rvert)$ is defined in the unshaded region, between the two space-time hyperboloids.
• Figure 4: Solutions to the ODE initial value problem \ref{['eq:DuffingIVP']} in the $(X, E)$ plane. For $E<1/4$, the solutions may "bounce" on the boundary of $\mathop{\mathrm{Ran}}\nolimits(\Phi)$, given by $E=\frac{1}{2} X^2-\frac{1}{4} X^4$.
• Figure 5: The maximal positive time $T_+(X, E)$ equals $S$, $R$ or $\infty$ depending on the location of $(X, E)$. This picture depicts the $Y\ge 0$ case.

Theorems & Definitions (41)

• Theorem 1
• Remark 1.1
• Remark 1.2
• Theorem 2: Self-similar behaviour inside the wave cone
• Theorem 3
• Conjecture 1
• Conjecture 2
• Theorem 4: Asymptotics of Lebesgue and Sobolev norms at the threshold
• Remark 1.3
• Conjecture 3