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Almost global existence means a lifespan which grows exponentially in the inverse size of small initial data, typically $T_{\epsilon }\ge \exp (c/\epsilon )-1$ for derivative-quadratic semilinear wave equations in three dimensions. Commuted wave energy and the Klainerman-Sobolev inequality lead to a logarithmic accumulation $\epsilon \log (1+T)$ in a bootstrap argument. This implies existence up to every fixed inverse power ${\epsilon }^{-N}$ when the data are small enough depending on $N$.

A first-order vector field $Z$ is a commutation field for the flat wave equation when $[\square ,Z]$ is zero or a controlled multiple of $\square$. Translations, spatial rotations and Lorentz boost vector fields commute with $\square$; the scaling vector field satisfies $[\square ,S]=2\square$. Repeated commutation preserves the differential order of a semilinear wave equation.

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.

Small data global regularity for wave maps holds in four spatial dimensions. Smallness is measured in sufficiently high weighted Sobolev norms relative to a constant map, with localized compatible Cauchy data. The vector field method for wave equations gives derivative decay $(1+t{)}^{-3/2}$. This is time-integrable, so commuted wave energy estimates close a small-data bootstrap argument for the derivative-quadratic semilinear wave equation. Higher regularity persists. No smallness of energy alone is asserted.

By the contraction mapping theorem, the contraction $A:X_{b,\tau }\to X_{b,\tau }$ has a unique fixed point $u$. The identity $Au=u$ says exactly that $u$ is the weak solution of the linear initial boundary value problem whose forcing is $u^2\sin u$. Consequently $u\in X_{\tau }$ satisfieswith the prescribed initial and homogeneous Dirichlet data. Thus a sufficiently small $\tau >0$ gives a local weak solution of the semilinear wave equation, as summarized by the local weak solution by contraction for a semilinear wave equation.

For a smooth semilinear wave equation whose nonlinearity is a smooth function of the solution and its first derivatives, local Sobolev space theory gives a continuation criterion: a finite maximal existence time requires divergence of a sufficiently high solution norm. A bounded base norm $H^{s+1}\times H^s$, with $s>n/2$, allows restarting local existence. Higher regularity persists by differentiated energy estimates. Precise norms can be weakened for particular nonlinearities.

For an integer $k\ge 0$ and sufficiently regular bounded functions, $\Vert fg{\Vert }_{H^k}\le C_k(\Vert f{\Vert }_{\infty }\Vert g{\Vert }_{H^k}+\Vert g{\Vert }_{\infty }\Vert f{\Vert }_{H^k})$. The Leibniz rule and derivative interpolation give the estimate. In a differentiated semilinear wave equation, this separates a controlled low-order coefficient from an arbitrarily high Sobolev norm, allowing propagation of smoothness after a fixed base norm has been controlled.

For a real scalar wave equation, the kinetic and gradient energy is $E_{\mathrm{lin}}(t)=\frac{1}{2}\int (|u_t{|}^2+|\nabla u{|}^2)dx$. A potential term is added for a semilinear wave equation $u_{tt}-\Delta u+V^{\prime }(u)=0$, giving the conserved wave energy $E=E_{\mathrm{lin}}+\int V(u)dx$.