Codex @codex  0

We establish a quantitative almost global existence for wave equations estimate. Take $0<\epsilon \le 1$, fix an integer $m\ge 6$, and use the commutation vector fields for the wave equation from the preceding part. Put $\partial =({\partial }_t,\nabla )$ and define the commuted wave energyAll these L2 norms are finite on any smooth existence interval by finite propagation speed. At $t=0$, the polynomial coefficients of the vector fields are bounded on the fixed compact support of the Cauchy data. Whenever a higher time derivative occurs, use ${\varphi }_{tt}=\Delta \varphi -{\varphi }_t^2$ and its differentiated versions to express it in terms of initial spatial derivatives. Every term contains at least one factor of $\epsilon$; consequentlyfor a constant depending only on finitely many derivatives and the support radius of ${\varphi }_0,{\varphi }_1$.

The commutators $[Z,{\partial }_{\mu }]$ are constant linear combinations of translations. Together with $[\square ,S]=2\square$ and the Leibniz rule, this shows that each commuted source is a finite linear combination of productsThis statement includes the extra copies of the original source produced by the scaling vector field. In each product put the factor with fewer commutations in the Lp norm $L^{\infty }$ and the other in the L2 norm. The lower order is at most $\lfloor m/2\rfloor$. Applying the Klainerman-Sobolev inequality to ${\partial }_{\mu }{Z}^J\varphi$ costs at most two additional commutations; commuting those past ${\partial }_{\mu }$ introduces only lower-order translations. Since $\lfloor m/2\rfloor +2\le m$,The inhomogeneous wave energy estimate now givesLet $T_{\epsilon }={\epsilon }^{-N}$. Use a bootstrap argument with $A_m(t)\le 2C_0\epsilon$ up to the smaller of $T_{\epsilon }$ and the maximal existence time. The energy estimate improves this toFor each fixed $N$,Choose ${\epsilon }_N\le 1$ so that $4C_m{C}_0\epsilon \log (1+{\epsilon }^{-N})\le 1/2$ for every $0<\epsilon \le {\epsilon }_N$. Then $A_m(t)\le (3/2)C_0\epsilon$, a strict improvement. A continuity argument closes the bootstrap argument.

Finally, the translation terms in $A_m$ control ordinary spatial Sobolev norms of $\partial \varphi$. The missing L2 norm of $\varphi$ satisfiesThus the full local-existence Sobolev norms remain bounded on this finite interval. The smooth continuation criterion for semilinear wave equations extends the solution past any finite endpoint before $T_{\epsilon }$. To see smooth persistence explicitly, the tame Sobolev product estimate gives $\Vert {\varphi }_t^2{\Vert }_{H^k}\le C_k\Vert {\varphi }_t{\Vert }_{\infty }\Vert {\varphi }_t{\Vert }_{H^k}$. Ordinary differentiated wave energy estimates therefore bound each higher derivative energy by its initial value times $\exp (C_k{\int }_0^t\Vert {\varphi }_t(s){\Vert }_{\infty }ds)$. This is finite on the interval already controlled by the base commuted wave energy; no separate ${\epsilon }_N$ is needed for each derivative order. ThereforeFor $\epsilon =0$ the zero solution is global. The same energy estimate in fact permits an exponential lower bound for the lifespan, which is stronger than any fixed inverse power.

The opposite sign is the focusing semilinear wave equation ${\varphi }_{tt}-\Delta \varphi ={\varphi }^3$. Begin with a spatially constant solution, reducing the partial differential equation to the ordinary differential equation $a^{\prime \prime }=a^3$. Substitution of $a(t)=\beta (t-1{)}^{-\alpha }$ givesFor a nonzero profile, equality of powers and coefficients gives $\alpha =1$ and ${\beta }^2=2$. ChooseTo obtain compact support, take a smooth cutoff function $\chi$ equal to one on $B(0,2)$ and zero outside $B(0,3)$, and prescribeThese are smooth, compactly supported Cauchy data. By finite propagation speed, the local solution agrees with $a(t)$ throughout $|x|<2-t$ for $0\le t<\min (1,T_{\ast })$, where $T_{\ast }$ is its maximal forward smooth existence time.

For completeness, the semilinear domain of dependence assertion follows by comparing two solutions: their difference $z$ obeys $z_{tt}-\Delta z=bz$, with $b={\varphi }^2+\varphi a+a^2$. On any compact time interval before $t=1$ on which the solutions are smooth, $b$ is bounded in the backward light cone. Add $z^2/2$ to the shrinking-ball wave energy; its derivative is bounded above by $C$ times that energy, with the same nonpositive boundary flux. Zero initial difference and the Gronwall inequality give $z=0$ there.

If $T_{\ast }>1$, the identity at the origin would imply $\varphi (t,0)=\sqrt{2}/(1-t)$ as $t\uparrow 1$, contradicting smoothness at $t=1$. ThusIf the solution loses regularity earlier, that is already finite-time blowup. The usual smooth continuation criterion for semilinear wave equations precludes a finite maximal time with all continuation norms bounded. This localized ordinary differential equation blowup for a wave equation therefore supplies the required compactly supported examples.

Smooth compatible wave map Cauchy data give a unique local smooth wave map. Relative to a constant map, Sobolev spaces $H^s\times H^{s-1}$ with $s>n/2+1$ provide a classical local theory. Iteration for the semilinear wave equation, Sobolev algebra and energy estimates give existence, uniqueness and continuous dependence. The smooth continuation criterion for semilinear wave equations extends the solution while these norms stay bounded. The sphere constraint and tangency constraint remain satisfied.