# Global solutions of the wave equation with bounded initial condition

Let $$f,g$$ be bounded compactly supported smooth functions, and assume $$u$$ is the solutions of the wave equation $$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$ $$u(x,0)=f, \ \ u_t(x,0)=g, \ \ x\in \mathbb{R}^n,$$

where $$c(x)>c_0>0$$ is also a bounded smooth functions on $$\mathbb{R}^n.$$

Does $$u(x,t)$$ remain bounded on $$\mathbb{R}^n \times (0,\infty)$$? It seems to me that this should follow from a standard result about hyperbolic equations but I can't find a relevant reference.

The answer is no, already for the wave equation $$c(x)=1$$. Let me be more precise.
If the initial data belong to $$H^s\times H^{s-1}$$ with $$s$$ strictly larger than $$n/2$$, then of course energy estimates give you that the $$H^s$$ norm remains bounded and hence the solution remains bounded in $$L^\infty$$ for all finite times. You may still have that the sup norm is unbounded as $$t\to\infty$$, depending on the precise form of the equation; for the constant coefficient case you get a global bound.
If the initial data are only bounded and $$n\ge2$$ then one can construct bounded $$f$$ (take $$g=0$$) such that $$u$$ is unbounded as $$t\to1$$. Consider for instance $$n=3$$: by Kirchhoff's representation of the solution we get $$u(0,t)=c\int_{\partial B(0,t)}[f(y)+t\partial _r f(y)]\ dS(y)$$ where $$\partial_r$$ is the derivative in the radial direciion. If $$f$$ is bounded but the radial derivative is singular on the boundary of $$B(0,1)$$ you can easily make the integral go to infinity as $$t$$ approaches 1.
If $$n=1$$ the answer is positive also with variable coefficients since the conservation of the $$H^1$$ energy controls the sup norm. You may have blow up at infinity of the sup norm depending on the precise form of the equation; if it is in divergence form you get a global bound.
• Thank you Piero. How about the case when $1-c(x)$ is compactly supported? Can we also get a global bound for all time? Mar 19 '19 at 13:58
• At what rate the $H^s$ norm could grow in time if it doesn't remain bounded? Could the growth be exponential? Mar 19 '19 at 14:13