Semilinear wave equation

A semilinear wave equation has the formso its highest-order part is linear while the forcing may depend nonlinearly on the solution and its lower derivatives.

A wave map is a harmonic map with Lorentzian domain: a critical point of the action obtained by contracting the pullback target metric with the domain Lorentzian metric. For target sphere $S^2$ in Euclidean space and $\square =-{\partial }_t^2+\Delta$, its extrinsic equation is $\square \varphi =\varphi (|{\varphi }_t{|}^2-|\nabla \varphi {|}^2)$, subject to $|\varphi |=1$. The derivative contractions are null forms for wave equations. Wave map Cauchy data must include a tangent initial velocity.

Smooth compatible data sufficiently small in high weighted Sobolev norms relative to a constant map produce global smooth wave maps in three and four spatial dimensions. In four dimensions, derivative decay $(1+t{)}^{-3/2}$ is time-integrable and closes commuted wave energy estimates. In three dimensions the weaker $(1+t{)}^{-1}$ decay requires the cancellation of null forms for wave equations, exploited by the vector field method for wave equations. These classical localized-data statements do not assert global regularity from small supercritical energy alone.

For smooth localized compatible wave map Cauchy data and compact Riemannian target, one-dimensional wave maps stay smooth globally. In null coordinates, the equation is $D_v{\varphi }_u=D_u{\varphi }_v=0$. Compatibility of the target metric with the covariant derivative makes $|{\varphi }_u{|}^2$ independent of $v$ and $|{\varphi }_v{|}^2$ independent of $u$. These transported derivative bounds and higher energy estimates prevent finite-time breakdown.

A time-independent wave map is a harmonic map of its spatial domain into the target. For target $S^2$, its equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. In two dimensions inverse stereographic projection gives a smooth nonconstant example of finite wave map energy $4\pi$.

The conserved energy of a wave map from Minkowski spacetime to a Riemannian target is $E=\frac{1}{2}\int (|{\varphi }_t{|}^2+|\nabla \varphi {|}^2)dx$, with norms measured in the target metric. Contraction of the wave map equation with the velocity and integration by parts gives conservation.

Under ${\varphi }_{\lambda }(t,x)=\varphi (t/\lambda ,x/\lambda )$, the conserved wave map energy scales as ${\lambda }^{n-2}E$. Energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\ge 3$. The critical derivative index of the homogeneous Sobolev space is $s_c=n/2$. Small energy alone is not an appropriate general small-data regularity hypothesis in supercritical dimensions.

Initial position ${\varphi }_0$ for a wave map takes values in the target, and initial velocity ${\varphi }_1$ belongs to its tangent space at ${\varphi }_0(x)$. For $S^2$ this means $|{\varphi }_0|=1$ and ${\varphi }_0\cdot {\varphi }_1=0$. Localized data agree with a constant map and zero velocity outside a compact set; the map itself need not be a compactly supported ambient function.

A quadratic derivative expression is a null form when its symbol vanishes on parallel null covectors. The Lorentz contraction $Q_0(f,g)=-f_t{g}_t+\nabla f\cdot \nabla g$ is the basic example. Its cancellation suppresses interactions of derivatives pointing along the same light ray. It is the algebraic structure behind the classical null condition for wave equations.

For a quadratic derivative source $A^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u$, the classical null condition is $A^{\alpha \beta }{\ell }_{\alpha }{\ell }_{\beta }=0$ whenever $-{\ell }_0^2+|{\ell }_{\mathrm{sp}}{|}^2=0$. It removes the leading interaction of parallel lightlike derivatives. In three dimensions this structure gives global smooth solutions for sufficiently small, localized data through the vector field method for wave equations. This PDE condition is distinct from the null condition asserting that a single vector is lightlike.

The focusing power wave equation is $u_{tt}-\Delta u=u|u{|}^{p-1}$. Its conserved energy has the negative potential term $-|u{|}^{p+1}/(p+1)$ and is not positive definite. Localized ordinary differential equation blowup for a wave equation produces smooth compactly supported data with finite maximal lifespan for suitable focusing powers.

A spatially constant blowup profile for a wave equation solves an ordinary differential equation. Choose smooth compactly supported initial data which agree with the profile on a ball larger than its blowup time times the wave speed. Finite propagation speed preserves agreement in an interior light cone, forcing finite-time breakdown of the localized solution. For $u_{tt}-\Delta u=u^3$, the profile $u(t)=\sqrt{2}/(1-t)$ is one example.

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.

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 $N$.

The defocusing power-type wave equation has the conserved nonnegative energy

For $u_{tt}-\Delta u+u^3=0$ in three spatial dimensions, conserved positive wave energy $E$ controls $\Vert \nabla u{\Vert }_2$. The Sobolev inequality gives $\Vert u^2\nabla u{\Vert }_2\lesssim \Vert \nabla u{\Vert }_2^2\Vert D^{2}u{\Vert }_2\lesssim E\Vert D^{2}u{\Vert }_2$. Differentiated wave energy estimates and the Gronwall inequality then give $\Vert u(t){\Vert }_{{\dot{H}}^2}+\Vert u_t(t){\Vert }_{{\dot{H}}^1}\le CD{e}^{CEt}$, where $D$ bounds the corresponding initial Sobolev norms.

Multiplying a defocusing wave equation by $\nabla \psi \cdot \nabla u+(\Delta \psi )u/2$ and integrating by parts gives a weighted spacetime identity. For a radial solution and radial weight, its coercive terms are

In three dimensions, $\Delta |x|=2/|x|$ away from the origin and $-\Delta (1/|x|)=4\pi {\delta }_0$. Hence $-{\Delta }^2|x|=8\pi {\delta }_0$ as a distribution.

For a finite-energy defocusing power-wave solution in three dimensions,Hardy's inequality bounds the Morawetz action by the conserved energy.

For a radial function $u(t,r)$ in three dimensions, setting $w=ru$ removes the radial first derivative because $r\Delta u=w_{rr}$. The defocusing power equation becomeson the half-line.

In three dimensions set $w=ru$ for a radial function satisfying $\square u=F$. If the source has support in $s\le r\le s+1$ and $|F(s,r)|\le (1+s{)}^{-2}$, the Duhamel principle bounds its contribution to $w(t,t+a)$ by $\frac{1}{2}\log (1+t)$ for $1/2\le a\le 1$. Its backward characteristic interval meets the source in a set of length at most one, and $r|F|\le (1+s{)}^{-1}$. Division by $r\simeq 1+t$ gives $|u|\lesssim \log (2+t)/(1+t)$ for compactly supported initial data.

For $q=w_t+w_r$ and a radial weight $\psi$,

Suppose the inhomogeneous linear wave equation has an energy estimate in a Banach solution space $X_T$, and its nonlinearity obeystogether with the corresponding locally Lipschitz estimate. For a sufficiently large closed ball and sufficiently small $T$, the map that solves the linear equation with source $F(w)$ is a contraction. The contraction mapping theorem then gives a local weak solution.