Characteristic localization, sharp lifespan asymptotics and global dynamics for a one-dimensional derivative wave equation

Abstract

We study the Cauchy problem $$ v_{tt}-v_{xx}=\abs{v_t+v_x}^{m}\abs{v_t}^{n}, \qquad v(x,0)=ηφ(x),\quad v_t(x,0)=ηψ(x), $$ on $\mathbb{R}$, with compactly supported profiles $\phi \in C_0^2(\mathbb{R})$, $\psi \in C_0^1(\mathbb{R})$, exponents $m,n>1$, and amplitude $\eta >0$. We show that the sign of $P_0=\psi +{\phi }^{\prime }$ decides the behaviour of small solutions, whatever the sign of $\psi -{\phi }^{\prime }$. If $P_0$ is positive at some point, the lifespan $T(\eta )$ obeys explicit two-sided bounds of order ${\eta }^{-(m+n-1)}$ for every $\eta >0$, and $$\underset{\eta \to 0^{+}}{\lim }{\eta }^{m+n-1}T(\eta )=\frac{2^n}{(m+n-1)({\max }_{\mathbb{R}}{P}_0{)}^{m+n-1}}.$$ If $P_0\le 0$, the solution is global for every amplitude below an explicit threshold. If moreover $P_0\not\equiv 0$, the component $v_t+v_x$ decays at the universal rate $(2^n/((m+n-1)t){)}^{1/(m+n-1)}$, and the gradient of the solution converges uniformly to that of a free wave travelling to the right; if $P_0\equiv 0$, the solution is itself such a travelling wave. The analysis rests on a localization property: the zero set of $v_t+v_x$ is invariant along its own characteristics, so that the nonlinear source stays in a slab of fixed width moving with speed one, and each characteristic of the other family is forced only during a bounded time. This yields the global existence, the long-time behaviour and the exact value of the limit. The results extend to the endpoint exponents $m,n\ge 1$ and to sources $f(v_t+v_x)g(v_t)$, for which $T(\eta )$ is asymptotic to an Osgood-type integral.

arXiv comment22 pages, no figures. Substantially revised and expanded version. The one-sided condition on the second characteristic component has been removed. Exact small-data lifespan asymptotics, a global-existence alternative and long-time asymptotics are established. The results are also extended to endpoint exponents and to a class of general nonlinear sources. Title changed