Bounds on Solution Size

Under conservative conditions, the bounds (6.19) hold, regardless of whether ${\mathcal D}$, the spatial domain, is defined as the entire $x$ axis, a semi-infinite domain, or a finite interval. It is important to note that both such bounds apply to derivatives of the dependent variable, and not the dependent variable itself. This is in direct contrast to the case of the harmonic oscillator, and might seem counterintuitive, but follows directly from the definition of the 1D wave equation itself: note that second-derivatives only appear, so that any solution of the form

$$u(x,t) = a_{00} + a_{01}t + a_{10}x + a_{11}xt$$ (6.27)

automatically satisfies the wave equation for any constants $a_{00}$, $a_{10}$, $a_{01}$ and $a_{11}$, and such a solution can clearly not be bounded for all $t$ and all $x$. The wave equation, unless properly terminated, allows a solution which is capable of drifting.

Nevertheless, conditions may indeed be employed to derive bounds of a less strict type on the size of the solution itself. Take, for instance, instance, the first of conditions (6.19). One may write

$$\Vert u\Vert _{{\mathcal D}}\frac{d}{dt}\Vert u\Vert _{{\mathcal ... ...t}\Vert _{{\mathcal D}}\leq \Vert u\Vert _{{\mathcal D}}\sqrt{2{\mathcal H}(0)}$$ (6.28)

where the first inequality above follows from the Cauchy-Schwartz inequality, and the second is results from the first of bounds (6.19). One thus has

$$\frac{d}{dt}\Vert u\Vert _{{\mathcal D}}\leq \sqrt{2{\mathcal H}(... ..._{{\mathcal D}}(t)\leq \Vert u\Vert _{{\mathcal D}}(0)+\sqrt{2{\mathcal H}(0)}t$$ (6.29)

and the norm of the solution at time $t$ is bounded by an affine function of time $t$; growth is no faster than linear.

As one might expect, better bounds are possible if fixed boundary conditions are employed. Considering again the wave equation defined over the semi-infinite domain ${\mathcal D} = {\mathbb{R}}^{+}$, with the Dirichlet condition (6.20) applied at $x=0$, one has, at any point $x=x_{0}$,

$$\vert u(x_{0},t)\vert = \vert\int_{x=0}^{x=x_{0}}u_{x}(x,t)dx'\ve... ... D}}\Vert 1\Vert _{[0,x_{0}]} \leq \frac{1}{\gamma}\sqrt{2{\mathcal H}(0)x_{0}}$$ (6.30)

The magnitude of the solution at any point $x_{0}$ in the semi-infinite domain may thus be bounded in terms of its distance from the end point. Notice that this is in fact a much stronger condition than a bound on the norm of the solution (in fact, it is such a bound, but in a Chebyshev, or $L^{\infty}$ type norm. If the spatial domain ${\mathcal D}$ above is changed to the finite interval ${\mathcal D}=[0,1]$, the above analysis is unchanged, and, as long as the boundary condition at $x=1$ is conservative, one may go further and write

$$\vert u(x_{0},t)\vert \leq \frac{1}{\gamma}\sqrt{2{\mathcal H}(0)... ...\qquad\Vert u\Vert _{{\mathcal D}} \leq \frac{1}{\gamma}\sqrt{2{\mathcal H}(0)}$$ (6.31)

which is indeed a bound on the $L^{2}$ norm of the solution. If the boundary condition at $x=1$ is of Dirichlet type, an improved bound is possible. See Problem 6.1.