The parents of a brain-damaged boy at the centre of a life-support treatment fight want a High Court judge to delay a decision to give them time to gather more evidence.
Specialists at King's College Hospital in London say giving further intensive care treatment to 11-month-old Isaiah Haastrup is "futile, burdensome and not in his best interests".
Bosses at King's College Hospital NHS Foundation Trust have asked Mr Justice MacDonald to let doctors stop providing life-support treatment and allow Isaiah to die.
Isaiah's mother, Takesha Thomas, and father, Lanre Haastrup, who are both 36 and from Peckham, south-east London, want treatment to continue.
Mr Justice MacDonald has overseen a trial in the Family Division of the High Court in London. He finished analysing evidence on Wednesday.
Miss Thomas and Mr Haastrup say they think doctors in Europe might be able to offer care options for Isaiah and have asked for time to carry out research.
The judge said he was aiming to produce a ruling next week but would consider their request.