I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side: $$ \Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial t}+\beta^2w\right)+\gamma^2\frac{\partial^2 w}{\partial t^2}=P\left[\theta(t)-\theta(t-\tau)\right]\delta(x-x_0(t))\delta(y-y_0(t)),~~ (x,y,t)\in(-l_1,l_1)\times(-l_2,l_2)\times(0,T), $$ subject to $$ w=\frac{\partial^2 w}{\partial x^2}=0,~~ x=-l_1; l_1,~~ {\rm for~ all}~ (y,t)\in[-l_2,l_2]\times[0,T], $$ $$ w=\frac{\partial^2 w}{\partial y^2}=0,~~ y=-l_2; l_2,~~ {\rm for~ all}~ (x,t)\in[-l_1,l_1]\times[0,T], $$ and $$ w(x,y,0)=w_0(x,y),~~ \frac{\partial w}{\partial t}\bigg|_{t=0}=w_0^1(x,y),~~ {\rm for~ all}~ (x,y)\in[-l_1,l_1]\times[-l_2,l_2]. $$

Here $\Delta$ is the Laplacian, $\theta(t)$ is the Heaviside`s function, $\delta(t)$ is that of Dirac, $u\in L^\infty[-l_1,l_1]\times[-l_2,l_2]$, $x_0$, $y_0$ are continuous, $w_0$ and $w_0^1$ are, at least, piecewise continuous and in consistency with boundary conditions, $\tau<T$.

Before starting to find the coefficient $u$, a question arose: in which Sobolev space the problem is well posed?

The energy of the system without damping term is $$ E(t)=\int_{[-l_1,l_1]\times[-l_2,l_2]}\left[\left(\Delta w\right)^2+\gamma^2\left(\frac{\partial w}{\partial t}\right)^2\right]dxdy, $$ so the starting point is that $w\in H^2\left([-l_1,l_1]\times[-l_2,l_2];W^{2,1}[0,T]\right)$, but only intuitively.