Prime Minister Julia Gillard flew to welcome Indonesian President Susilo Bambang Yudhoyono to Darwin yesterday as winter settled on her fight for political survival.

The second of their now-annual talks will focus heavily on asylum seekers, one of the heaviest weights around Gillard's neck and one for which she now desperately seeks answers after the collapse last week of attempts to reach a compromise deal with the Opposition.

Australia yesterday said it would give four early-model Hercules transport aircraft to Indonesia, amid warming relations that have seen several goodwill gestures from both sides.

But Yudhoyono's visit is unlikely to produce any real measures to ease Gillard's load: asylum seekers have a much lower political profile in Indonesia, which regards the issue as a regional problem requiring co-ordinated action from countries of origin, transit and destination.

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And there is a fiercer domestic demon stalking Gillard. Her carbon tax came into force on Sunday, unleashing a fury of political activity almost as frantic as an election campaign.

While Gillard flew north after a blitz of breakfast radio and television shows, her ministers continued to lock horns with Opposition leader Tony Abbott and his colleagues over the tax.

Polls yesterday showed Abbott winning decisively. Gillard's hopes of Labor's stocks rising phoenix-like as the tax beds down appear to be slim.

The carbon tax is about as popular as former Liberal Prime Minister John Howard's GST after its introduction - and Howard, a much more popular leader of a far stronger Government, took 18 months or so to fully recover.

Last month's annual Lowy poll showed a dramatic cooling in the importance Australian voters attach to tough action on climate change, with 63 per cent opposing the carbon tax, 45 per cent of them strongly.

A Nielsen poll in the Age yesterday confirmed the trend, reporting that opposition to the tax had risen three points in the past month to 62 per cent, including one-third of Labor voters and nine out of 10 Coalition supporters.

The Government says millions of households will be better off with compensation that has already started flowing and new tax cuts that also came into force this week. In the coming financial year average household assistance to offset cost of living rises will run to about A$10 ($12.76).

Voters remain unconvinced. The Nielsen poll said 51 per cent believed they would be worse off with the tax, although 37 per cent thought it would not make a difference and 5 per cent said they would be better off.

The news on voting intentions was even worse. Nielsen said the Coalition's two-party preferred lead had widened to 58-42 per cent, sufficient to produce an 8 per cent swing that would bury Labor.

In the Australian, a Newspoll said Labor's support in Queensland had fallen so low the party faced a potential 10 per cent swing to the Coalition that would leave the state without a single Labor MP - including former Prime Minister Kevin Rudd, still hovering in the background as Gillard's grip on the leadership becomes more shaky.

Gillard did receive one welcome boost yesterday, with 300 companies belonging to Businesses for Clean Energy - including Westpac, Unilever, GE and AGL - signing a statement supporting the tax as a key move for future growth.

She said Australians would now have the chance to judge carbon pricing for themselves, beyond the "doomsday merchants' ... hysterical fear campaign".

"We've been through these kind of cycles in the past with big reforms that have been very controversial at the time, that now when people look back on them, people can't imagine that there was even ever a debate," she said.

But Abbot repeated his pledge to immediately repeal the tax if he won power.

"On the first day, I will issue instructions to draft legislation that repeals the carbon tax," he said. "Within the first month, the Cabinet will approve legislation to repeal the carbon tax and on the first sitting day, the new Government will introduce legislation to repeal the carbon tax."

The Labor Government was likely to get some good news late last night with the expected release in Libya of Australian lawyer Melinda Taylor.

Foreign Minister Bob Carr said Taylor would be released from custody three and a half weeks after she and three other International Criminal Court officials were detained in Zintan.

Taylor would then be taken to Tripoli to board a flight bound for Europe. She was expected to be reunited with husband Geoff and 2-year-old daughter Yasmina in The Hague today.

"So far it's very good news, a great relief," Carr told Sky News. Taylor's release was a result of "fruitful" talks between Libyan authorities and the ICC. Carr has also lobbied hard for her release, even visiting the Libyan capital Tripoli last month to press her case.

Taylor was detained while representing Saif al-Islam, the son of former dictator Muammar Gaddafi. She was found carrying documents that were judged a threat to national security.

- additional reporting AAP