Here is one possible approach.

Choose smooth approximations of $H$. For example consider a smooth function

$$ \eta:\mathbb{R}\to [0,\infty)$$

such that

$$\eta (t)=0,\;\;\forall t\leq 0$$

$$\eta(t)=1,\;\;\forall t\geq 1.$$

Now define

$$H_\varepsilon(t)= \eta(t/\varepsilon)$$

Then

$$\lim_{\varepsilon\searrow 0} H_\varepsilon (t) =H(t),\;\;\forall t\neq 0$$.

Now investigate the above equation with $H$ replaced with $H_\varepsilon$ an hope that you understand what happens when $\varepsilon \searrow 0$.

On the other hand, there is a large class of evolution equations with discountinuous right-hand side described by the so called maximally monotone operators. For 2nd order equations such as yours, one of the most influential papers is

Barbu, Viorel Existence theorems for a
class of two point boundary problems.
J. Differential Equations 17 (1975),
236–257.

For the general theory of evolution equations involving maximal monotone operators the best sources are

Barbu, Viorel Nonlinear
differential equations of monotone
types in Banach spaces. Springer
Monographs in Mathematics. Springer,
New York, 2010. x+272 pp.

Brezis, Haim Opérateurs maximaux
monotones et semi-groupes de
contractions dans les espaces de
Hilbert. (French) North-Holland
Mathematics Studies, No. 5. Notas de
Matemática (50). North-Holland
Publishing Co., Amsterdam-London;
American Elsevier Publishing Co.,
Inc., New York, 1973. vi+183 pp