Clearly your TV has a fault in the circuitry that delivers the picture - it seems to be dependant on how long the set has been on for - a heat related fault - it might be a failing component or a poor joint (which amounts to the same thing).
Sometimes a confident amateur can track down a fault like this with controlled tapping and/or freezer spray. But it really needs test equipment - circuit schematics and experience.
It will certainly be repairable but only you can judge the likely cost of repair versus the cost of a new TV
Amazon offer a 19" TV with built in DVD for under £90