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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.

We prove finite propagation speed using a shrinking cone energy argument. Fix $T>0$ and $x_{\ast }$, and suppose the Cauchy data vanish on $B(x_{\ast },T)$. For $0\le s<T$, define the local wave energyThe homogeneous wave equation gives the local conservation lawDifferentiate the integral over the moving ball. Its boundary moves inward with speed one, so the divergence theorem givesHere $n$ is the outward unit normal, ${\partial }_n$ the normal derivative, and ${\nabla }_{\tan }$ the component of the gradient tangent to the boundary. Since $E(0)=0$ and $e\ge 0$, we have $E(s)=0$. Hence both ${\varphi }_t$ and $\nabla \varphi$ vanish inside the backward light cone. Integrating ${\varphi }_t(s,x_{\ast })=0$ from the zero initial displacement gives $\varphi (s,x_{\ast })=0$; continuity then gives $\varphi (T,x_{\ast })=0$.

Applying the same energy estimate to the difference of two solutions proves the domain of dependence assertion. If $\mathrm{dist}(x_{\ast },K)>T$, the initial ball misses $K$, and the preceding argument proves the stated support bound. No disturbance propagates faster than one.

The sign convention makes this a defocusing semilinear wave equation:Its wave energy includes a nonnegative potential term:For a smooth function solution, finite propagation speed preserves compact support on finite time intervals. Differentiate under the integral and use integration by parts:Equivalently the local conservation law has density $e={\varphi }_t^2/2+|\nabla \varphi {|}^2/2+{\varphi }^4/4$ and flux $-{\varphi }_t\nabla \varphi$. All three terms are nonnegative, and $E=0$ forces both Cauchy data to vanish. Hence this is a positive conserved energy, rather than the indefinite energy associated with the opposite sign.

We use the homogeneous version of the Sobolev embedding theorem in three dimensions:Here homogeneous Sobolev space ${\dot{H}}^1$ is the completion of compactly supported smooth functions in the L2 norm of the gradient, identified with its $L^6$ representative. Apply this Sobolev inequality both to $\varphi$ and to its spatial derivatives.

LetThe Plancherel theorem identifies the L2 norm of the Hessian matrix with $\Vert \varphi {\Vert }_{{\dot{H}}^2}$, because ${\sum }_{i,j}{\xi }_i^2{\xi }_j^2=|\xi {|}^4$. In particular,Differentiating the defocusing semilinear wave equation gives $\square ({\partial }_i\varphi )=3{\varphi }^2{\partial }_i\varphi$. By the Holder inequality with exponents $3$ and $6$, the preceding Sobolev inequality, and conservation of the positive wave energy,Use the inhomogeneous wave energy estimate simultaneously for the three spatial derivatives. It givesThe Gronwall inequality therefore yields $B(t)\le D\exp (CEt)$. We may take the continuous, locally bounded functionThe constant is universal; the dependence on the initial data is only through $D$ and $E$. The H2 bound for the defocusing cubic wave equation holds on every existing smooth interval, without assuming the global conclusion. If $D=0$ or $E=0$, the zero solution satisfies the same estimate.

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.

No: the first moment generally depends on the transverse coordinate. At entry to the random medium, the initial data are deterministic, sowhich is a nonconstant Gaussian beam profile. Statistical homogeneity of the medium says that shifting both the medium and the incident data gives correspondingly shifted field statistics. It does not make the response to fixed, localized incident data translation invariant.

In particular, Gaussian coherent-field propagation in the Markov approximation, derived in part (d), givesfor the linear weak-index model. A homogeneous attenuation factor multiplies the varying beam profile. Thus spatially homogeneous medium statistics can coexist with a transversely inhomogeneous coherent field. The first moment is not the mean wave intensity, and disappearance of the coherent component is not itself disappearance of the total wave energy.

For the homogeneous speed-one wave equation, integrate its wave energy density over $B(x_{\ast },T-t)$. The moving boundary contributes $-e$, while the usual flux contributes $u_t{\partial }_{n}u$. Their sum is $-\frac{1}{2}(u_t-{\partial }_{n}u{)}^2-\frac{1}{2}|{\nabla }_{\tan }u{|}^2\le 0$. Zero Cauchy data in the initial ball therefore force zero solution in its backward light cone.

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 $E=E_{\mathrm{lin}}+\int V(u)dx$.