Sailplane pilot reported a NMAC event during takeoff with a non-communicating plane departing on an intersecting runway. The sailplane pilot executed an evasive maneuver to avoid a collision.

Narrative

ZZZ has two intersecting runways. In light wind conditions both runways are in use for operations of powered glider; seaplane; and light aircraft cross-wind instruction. I monitored CTAF until sure no traffic would use XX/XY; announced my intentions; and taxied onto the runway for departure XX. A Piper Cub used for commercial rides and training announced base XZ. I announced I would hold until he was clear and he acknowledged. The cub landed and announced clear the active. I announced 'ZZZ Traffic motor-glider departing XX'. No other traffic announced on the radio during this period. I rolled; lifted off; and as I started to climb; I saw a plane taking off from XZ on a collision course over the intersection. At an altitude of 18m and climbing; I made a sharp right turn to avoid a collision. I estimate we came within 300 ft.I have been unable to identify the other aircraft. I thought it was an Icon A3 used for flight instruction at [a local flight school]; but their records seem to show it was in flight away from the airport at that time; and I am not sure.FlightRadar24 shows my flight (Aircraft X; with very low fidelity); but no other aircraft. Possibly the other aircraft's ADS-B was USA-only-978MHz which FlightRadar24 does not track?Suggestion: As this is a repeated problem at ZZZ; maybe signs by the thresholds? 'WARNING: Crossing Runway is Active!'