AMNA NAWAZ: Today, Russian President Vladimir Putin said Ukraine would have to stay out of NATO for there to be peace, while Ukrainian President Volodymyr Zelenskyy said he wouldn't agree to any cease-fire unless it included security guarantees for Ukraine.

During the just-ended campaign, candidate Trump said he wanted the war to end and that he could negotiate a quick resolution.

Nick Schifrin has the latest on the state of the war and what might come next as President Trump returns.

NICK SCHIFRIN: In Southern Ukraine today, an apartment reduced to rubble, and a frantic search for survivors, children carried across what used to be their home.

A Russian strike here killed at least four as Russian rockets pound Ukraine's 600-mile front.

Russians are making gains, capturing partially destroyed villages, raising the Russian tricolored flag.

MICHAEL KOFMAN, Carnegie Endowment for International Peace: I'm very concerned about the trajectory of this war.

I think it's quite clearly negative.

NICK SCHIFRIN: Michael Kofman is a Carnegie Endowment senior fellow and just returned from Ukraine.

MICHAEL KOFMAN: The Ukrainian military has been losing territory at an accelerated pace, particularly over the past three months.

Each month, Russia has been taking more territory relative to the last month.

NICK SCHIFRIN: Ukraine's biggest concern is in Donetsk, where Russian forces have focused and are threatening the key transit hub of Pokrovsk.

South of the city, Russian videos show Russian drones and bombs hunting Ukrainian soldiers and trying to encircle Pokrovsk.

The Russians are taking tremendous losses.

U.S. officials say more than 1,200 are killed or wounded every day.

But in parts of Donetsk, Ukrainian soldiers are outgunned and outmanned.

MICHAEL KOFMAN: Its forces are stretched along a very broad front, and if Russian units are able to get around Pokrovsk, they can then push further into other regions.

Russia is leveraging its overall advantage in manpower.

And despite the fact that Russian forces are taking very heavy losses, they are still replacing their losses, and they simply have a lot more manpower to throw at the problem.

NICK SCHIFRIN: Ukraine has tried to seize its own initiative to the north, taking territory this summer in Russia's Kursk.

Russian forces have managed to recapture about 30 percent of Kursk that Ukraine seized.

And now, Ukraine says 11,000 North Korean soldiers have joined the fight, although it's unclear if they're Russia's cavalry or cannon fodder.

VOLODYMYR ZELENSKYY, Ukrainian President: North Korea is now, in effect, waging war in Europe.

North Korean soldiers are attempting to kill our people on European soil.

NICK SCHIFRIN: But today in Budapest, Ukrainian President Volodymyr Zelenskyy and all of Europe's leaders had another country on their minds.

This was Europe's first summit since Donald Trump was reelected.

DONALD TRUMP, Former President of the United States (R) and Current U.S. President-Elect: I want the war to stop.

I want to save lives.

NICK SCHIFRIN: Trump's personal policy for Ukraine is unclear.

But one plan proposed by former Trump advisers calls for the U.S. to indefinitely delay Ukraine's NATO membership, continue to arm Ukraine, but condition further aid on Ukraine talking peace, offer Russia sanctions relief, and Ukraine would at least temporarily give up the right to use its military to recapture occupied territory.

SEN. J.D.

VANCE (R-OH), Vice President-Elect: What it probably looks like is something like the current line of demarcation between Russia and Ukraine.

That becomes like a demilitarized zone.

It's heavily fortified, so the Russians don't invade again.

Ukraine remains its independent sovereignty.

Russia gets the guarantee of neutrality.

NICK SCHIFRIN: Today, Putin said he's open to Trump's ideas.

VLADIMIR PUTIN, Russian President (through translator): What was said concerning the desire to restore relations with Russia to help end the Ukrainian crisis, in my opinion, seems to me to be at least worthy of attention.

NICK SCHIFRIN: But Zelenskyy said any deal requires guarantees of Ukraine's security.

VOLODYMYR ZELENSKYY (through translator): A cease-fire when there are no security guarantees, clear guarantees, believable guarantees, not some rhetorical ones, a cease-fire without them is a preparation for the continuation of the occupation, ruining our independence and sovereignty.

That's the cease-fire talked about today.

JOHN HERBST, Former U.S.

Ambassador to Ukraine: Zelenskyy recognized that he needs to work with the Trump administration, so he's signaled a readiness to get involved along these lines, but laying out also that what's important is a just peace.

NICK SCHIFRIN: John Herbst is the former U.S. ambassador to Ukraine.

He says Zelenskyy has told Trump what Ukraine needs and created a good relationship with Trump, but it's not clear what Trump will do.

JOHN HERBST: There are two contending parties within the Trump campaign.

One would call for major Ukrainian concessions and no guarantees.

The other calls for negotiations, but recognizes that making sure Putin does not take control of Ukraine is critical for American interests.

And that group in the Trump world is not intimidated by Russia's nuclear threats, the way the Biden administration has been.

NICK SCHIFRIN: If the U.S. were to reduce its military support to Ukraine, what would be the impact?

MICHAEL KOFMAN: Well, I think the impact would be very significant on Ukrainian military.

I don't think it would be immediate, but it would follow shortly thereafter.

I don't believe the European countries right now are in a good position to effectively substitute for everything the United States provides Ukraine.

And we're not talking about just material support in terms of ammunition, equipment.

We're also talking about training for Ukrainian forces, intelligence support and all sorts of things that people don't necessarily see or hear about as much.

NICK SCHIFRIN: And so Ukraine will have to make the case to Trump for more support as it struggles to hold the line.

For the "PBS News Hour," I'm Nick Schifrin.