= Uniqueness of Sobolev weak solutions of a wave equation
For the <weak energy solution of a variable-coefficient wave equation>, uniqueness can be proved without using $u_t$ as a spatial $H_0^1$ test function. For a difference with zero data, fix $s$ and test by $v(t)=\int_t^su(r)dr$ for $t<s$, extended by zero afterwards. Put $z(t)=\int_0^tu(r)dr$, so $v(t)=z(s)-z(t)$. <Integration by parts> in time for the symmetric principal form, and in space for the first-order terms, give
$$
\|u(s)\|_2^2+\theta\|Dz(s)\|_2^2
\leq C\int_0^s\bigl(\|u(t)\|_2^2+\|Dz(t)\|_2^2\bigr)dt+Cs\|Dz(s)\|_2^2.
$$
For short enough intervals the last term is absorbed. The <Gronwall inequality> forces $u=0$, and repetition proves uniqueness on the full interval. This antiderivative test is valid at the stated space-time $H^1$ regularity, unlike an unqualified direct energy test by $u_t$.
Back to article page
