I am interested in Airy's equation
$$\frac{\partial u}{\partial t}(t,x)=-\frac{\partial^3 u}{\partial x^3}(t,x)$$
on a bounded or semi-bounded domain, e.g. on $(-\infty,0)$. In order to obtain a group of isometries as propagator in $L^2(-\infty,0)$, two boundary conditions have to imposed - e.g., $u'(0)=0$ and either $u(0)=0$ or $u''(0)=0$. But then it seems that the spectrum of the operator is far too large, so I doubt that this operator does not generate a semigroup in $L^2(-\infty,0)$ at all, regardless of the imposed boundary conditions. Is it so? Has anybody ever proved well-posedness of the above equation in some reasonable function or distribution space?



Airy's equation on $\mathbb R_-$
The operators $A=-\,\partial\_x^3$ with domain $\mathscr D(A) =
\left(H^3\cap H\_0^2\right)(\mathbb R\_-)$ and $A^\ast=\partial\_x^3$
with domain $\mathscr D(A^\ast) = \left(H^3\cap H\_0^1\right)(\mathbb
R\_-)$ are $m$-dissipative on $L^2(\mathbb R\_-)$. This follows from
$\Re\langle Au,u\rangle =0$ for $u\in\mathscr D(A)$ and $\Re\langle
A^\ast v,v\rangle =-\,\frac12\,|v'(0)|^2\leq 0$ for $v\in\mathscr
D(A^\ast)$. Therefore, $A$ and $A^\ast$ generate $C\_0$-contraction
semigroups on $L^2(\mathbb R\_-)$, by the Lumer-Phillips
theorem. Moreover, the semigroup $\{e^{tA}\}\_{t\geq0}$ consists of
isometries (though not of unitary operators for $t>0$, as $iA$ is not
selfadjoint, see Christian's comment above).


One can also directly solve the initial-boundary value problem
$$
\left\{ \enspace
\begin{aligned}
 & u\_t + u\_{xxx} = f(t,x), & (t,x)\in \mathbb R\_+ \times \mathbb R\_-, \\
 & u\bigr|\_{x=0} = h\_0(t), \enspace u\_x\bigr|\_{x=0} = h\_1(t), \\
 & u\bigr|\_{t=0} = u\_0(x)
\end{aligned}
\right.
$$
(and similar for the operator $\partial\_t-\partial\_x^3$ with
one in place of two boundary conditions), e.g., by using the Laplace
transform with respect to $x$. See [N. Hayashi, E. Kaikina, Nonlinear
Theory of Pseudodifferential Equations on a Half-line, North-Holland,
2004,](https://www.elsevier.com/books/nonlinear-theory-of-pseudodifferential-equations-on-a-half-line/hayashi/978-0-444-51569-8) which contains a chapter on linear problems.


