Consider the equation:
$u_{t} - a \Delta u + \lambda u = g $ in $\mathbb R$ X $(0,T)$
$u(x, 0) = u_{0}(x)$
Where $a \gt 0$ & $\lambda \in \mathbb R$ are given constants!!
Now, assume that for some integer $m \geq 0$ $u_{0} \in H^{m}(\mathbb R)$ & $g \in L^{2}(0,T; H^{m-1}(\mathbb R))$
Then it is to be proved that: the solution $u$ of the above problem satisfies:
$u \in L^{2}(0,T;H^{m+1}(\mathbb R)) \bigcap \mathcal B(0,T;H^{m}(\mathbb R))$ ;
$u_{t} \in L^{2} (0,T;H^{m-1}(\mathbb R))$.
Notation: $\mathcal B(0,T;X) :=$ space of bounded continuous functions from $[0,T]$ to $X$
$L^{p} (0,T;X) := $ The space of functions $v : t \in (0,T) \to v(t) \in X $ which are strongly measurable w.r.t. the Lebesgue Measure $dt$ & satisfy:
$ ||v||_{L^{p} (0,T;X)} := [\int_{0}^{T}||v(t)||_{X}^{p}dt]^{1/p} \lt +\infty$ ; if $1\leq p \lt \infty$
$ ||v||_{L^{\infty} (0,T;X)} := ess sup_{t \in (0,T)}||v(t)||_{X} \lt \infty$