  We propose a fine analysis of second order optimality conditions for the
optimal control of semi-linear parabolic equations with respect to the initial
condition. More precisely, we investigate the following problem: maximise with
respect to $y\in L^\infty({(0;T)\times \Omega})$ the cost functional
$J(y)=\iint_{(0;T)\times \Omega}j_1(t,x,u)+\int_\Omega j_2(x,u(T,\cdot))$ where
$\partial_t u-\Delta u=f(t,x,u)+y\,, u(0,\cdot)=u_0$ with some classical
boundary conditions, under constraints of the form $-\kappa_0\leq y\leq
\kappa_1\text{ a.e.}\,, \int_\Omega y(t,\cdot)=V_0$. This class of problems
arises in several application fields. A challenging feature of these problems
is the study of the so-called abnormal set $ \{-\kappa_0<y^*<\kappa_1\}$ where
$y^*$ is an optimiser. This set is in general non-empty and it is important
(for instance for numerical applications) to understand the behaviour of $y^*$
in this set: which values can $ y^*$ take? In this paper, we introduce a
Laplace-type method to provide some answers to this question. This Laplace type
method is of independent interest.
