Apple has asked for a court order for a permanent US sales ban on Samsung Electronics products alleged to have violated its patents along with additional damages of $707 million on top of the billion-dollar verdict won by the iPhone maker last month.
Samsung has responded by asking for a new trial.
The world's top two smartphone makers are locked in patent battles in 10 countries as they vie for top spot in the lucrative, fast-growing market.
Apple scored a legal victory over Samsung in late August when a US jury found that the Korean firm had copied critical features of the iPhone and awarded the US firm $1.05 billion in damages.
In a motion filed late on Friday US time, Apple sought a further $400 million damage award for design infringement by Samsung; $135 million for willful infringement of its utility patents; $121 million in supplemental damages based on Samsung's product sales not covered in the jury's deliberation; and $50 million of prejudgment interest on damages through December 31. The requests together come to $707 million.
Apple wants the injunction to cover "any of the infringing products or any other product with a feature or features not more than colourably different from any of the infringing feature or features in any of the Infringing Products."
Such a wide-ranging sales ban could result in the extension of the injunction to cover Samsung's brand-new Galaxy S III smartphone.
'Rectangles with rounded corners'
Samsung, in a filing to the US court, asked for a new trial to be held.
"The Court's constraints on trial time, witnesses and exhibits were unprecedented for a patent case of this complexity and magnitude, and prevented Samsung from presenting a full and fair case in response to Apple's many claims," Samsung said.
"Samsung therefore respectfully requests that the Court grant a new trial enabling adequate time and even-handed treatment of the parties."
In a separate statement, Samsung lamented the fact that patent rulings should cover issues such as the shape of the product in addition to technological points.
"It is unfortunate that patent law can be manipulated to give one company a monopoly over rectangles with rounded corners, or technology that is being improved every day by Samsung and other companies," it said.
The Korean firm earlier this week said it plans to add Apple's new iPhone 5 to the existing US patent lawsuits, stepping up its legal challenge as the two companies seek to assert rights to key technologies.
Apple said it wanted the court to award it damages that reflect "a rational and fair effort to address Samsung's willful misconduct that has and will impose lasting harm on Apple."
The Korean firm was the world's top smartphone maker in the second quarter of this year, shipping more than 50 million phones, nearly double Apple's 26 million iPhone shipments.
Both companies are raising their marketing spending to promote their latest products ahead of the year-end shopping season.