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This paper investigates the initiation of a deflagration in a premixed boundary-layer stream by continuous heat deposition from a line energy source placed perpendicular to the flow on the wall surface, a planar flow configuration relevant for small-scale combustion applications, including portable rotary engines. Ignition is investigated in the constant density approximation with a one-step irreversible reaction with large activation energy adopted for the chemistry description. The ratio of the characteristic strain time, given by the inverse of the wall velocity gradient, to the characteristic deflagration residence time defines the relevant controlling Damköhler number $D$. The time-dependent evolution following the activation of the heat source is obtained by numerical integration of the energy and fuel conservation equations. For sufficiently small values of $D$, the solution is seen to evolve towards a steady flow in which the chemical reaction remains confined to a finite near-source reactive kernel. This becomes increasingly slender for increasing values of $D$, corresponding to smaller near-wall velocities, until a critical value $D_{c1}$ is reached at which the confined kernel is replaced by a steady anchored deflagration, assisted by the source heating rate, which develops indefinitely downstream. As the boundary-layer velocity gradient is further decreased, a second critical Damköhler number $D_{c2}>D_{c1}$ is reached at which the energy deposition results in a flashback deflagration propagating upstream against the incoming flow along the base of the boundary layer. The computations are extended to investigate the dependence of $D_{c1}$ and $D_{c2}$ on the fuel diffusivity as well as the additional dependence of $D_{c1}$ on the source heating rate, thereby delineating the boundaries that define the relevant regime diagram for these combustion systems.