Meanwhile, the, but they're not making much headway, reports CBS News correspondent Mark Phillips. Diplomats and European leaders raced around the region in search of a cease-fire, but with Palestinian rocket fire continuing, Israel said it won't stop its crippling 10-day assault until "peace and tranquility" are achieved in southern Israeli towns in the line of fire.
The operation's ground phase, which began Saturday with a withering round of artillery fire, was going according to plan, military spokesman Brig. Gen. Avi Benayahu told Israeli TV. Israeli forces were sweeping through Palestinian rocket launching locations near the border and the militants were suffering many casualties, he said.
In an interview at the Israel-Gaza border, Israeli military analyst Alon Ben-David told Phillips that the objectives of the operation include a sustainable cease-fire, thwarting smuggling from Egypt into Gaza, and "involvement of the Palestinian authorities in the Rafah crossing."
However, Ben-David cautioned that "we are not close" to the political goals.
Gaza health officials reported more than 550 Palestinian dead and about 2,500 wounded since Israel began the campaign 10 days ago, including 200 civilians. U.N. humanitarian chief John Holmes told reporters in New York on Monday that U.N. officials believe at least 500 people have been killed in the fighting and that as many as 25 percent are civilians.
At least 20 Palestinian children were killed during the day, said Dr. Moaiya Hassanain, a health official. Most confirmed deaths have been civilians.
The three young brothers from the Samouni family died in an attack on a town outside Gaza City, a Gaza health official said.
The voices from within Gaza have become more desperate, reports Phillips.
"We don't have enough food, we don't have enough water," said freelance journalist Sameh Habeeb as bombs rained down near him.
Five Israelis have been killed during the offensive, including a soldier in the ground operation. Heavy Israeli casualties could undermine what has so far been overwhelming public support for the operation.
The three brothers, reportedly killed in an artillery strike Monday, were carried to a cemetery in an emotional funeral. One of them, Issa Samouni, 3, was wrapped in a white cloth, showing only his pale, yellow face. A man delicately placed him in a dark grave cut into the earth.
In one of the first major gunbattles of the ground campaign, Israeli troops and Hamas militants clashed at close quarters on the outskirts of the crowded Gaza City neighborhood of Shajaiyeh, Israeli defense officials said.
Troops seized control of three six-story buildings on the outskirts, climbing to rooftop gun and observation positions. Residents were locked in their rooms and soldiers took away their cell phones, a neighbor said, quoting a relative who called before his phone was seized.
"The army is there, firing in all directions," said Mohammed Salmai, a 29-year-old truck driver. "All we can do is take clothes to each other to keep ourselves warm and pray to God that if we die, someone will find our bodies under the rubble."
Fighter jets attacked houses, weapons storage sites, a pair of mosques and smuggling tunnels, as they have since the start of the offensive Dec. 27. Israel has attacked several mosques during the campaign, saying they were used to store weapons.
In another strategic move, Israeli forces seized a main highway in Gaza, slicing the territory in two.
Details also emerged of an attempt by Hamas fighters to capture Israeli soldiers hours after the ground operation began. As infantrymen advanced up a strategic hill before dawn Sunday, militants emerged from a tunnel and tried to drag two Israeli infantrymen inside, according to Israeli defense officials.
The Israelis fought off the attempt, but one soldier was killed in an ensuing firefight. The officials spoke on condition of anonymity because the incident had not been made public.
Hamas already holds one Israeli soldier, captured in June 2006, and another would be an important bargaining chip.
Israeli forces detained 80 Palestinians - some of them suspected Hamas members - and transferred several to Israel for questioning, said military officials, speaking on condition of anonymity because they were not allowed to release the information.
The Gaza City area was rocked by shelling from both sides as gunboats in the sea and artillery and tanks closing in from the east unloaded thunderous fire.
After dark, the shelling reached deeper into residential areas. Fireballs lit up the horizon to the east, setting off blazes on the ground and silhouetting Gaza's tall buildings. Tracer fire ripped across the skyline.
Norwegian doctor Mads Gilbert is working alongside Palestinian surgeons in Gaza's largest hospital, in Shifa. In a telephone interview yesterday, he recounted the harrowing conditions and said anyone trying to depict this as a "clean war," was simply "a liar".
The State Department said the U.S. was pressing for a cease-fire that would include a halt to rocket attacks and an arrangement for reopening crossing points on the border with Israel, said spokesman Sean McCormack. A third element would address the tunnels into Gaza from Egypt through which Hamas has smuggled materials and arms.
President George W. Bush emphasized "Israel's desire to protect itself."
"The situation now taking place in Gaza was caused by Hamas," he said.
The deputy head of Hamas' politburo in Syria, Moussa Abu Marzouk, rejected the U.S. proposal, telling the AP the U.S. plan seeks to impose "a de facto situation" and encourages Israel to continue its attacks on Gaza.
French President Nicolas Sarkozy, who unsuccessfully proposed a two-day truce last week, met with Palestinian President Mahmoud Abbas, who lost control of Gaza to Hamas in June 2007.
Europe "wants a cease-fire as quickly as possible," Sarkozy said after meeting Abbas, urging Israel to halt the offensive, while blaming Hamas for acting "irresponsibly and unpardonably."
A European Union delegation met with Foreign Minister Tzipi Livni.
"The EU insists on a cease-fire at the earliest possible moment," said Karel Schwarzenberg, the foreign minister of the Czech Republic, which took over the EU's presidency last week from France. Rocket attacks on Israel also must stop, Schwarzenberg told a news conference with Livni.
The EU brought no truce proposals of its own because the cease-fire "must be concluded by the involved parties," he added.
The new Arab proposal at the U.N., which is still being formulated, calls for a cease-fire, an immediate withdrawal of Israeli troops, the opening of crossing points, monitoring mechanisms and a U.N. international peacekeeping force to be created for Gaza to protect the civilian population - but no mention was made about rocket attacks against Israel, reports CBS News foreign affairs analyst Pamela Falk.
As the bruising campaign ended its 10th day, Hamas pummeled southern Israel with more than 30 rockets and promised to wait for Israeli soldiers "in every street and every alleyway."
Israeli Defense Minister Ehud Barak said the offensive would go on until Israel achieved "peace and tranquility" for residents of southern Israel.
One of the rockets struck a large outdoor market that was closed at the time in the town of Sderot, just across Gaza's northeastern border. Another hit a kindergarten in the coastal city of Ashdod, north of the strip. The kindergarten, like schools across southern Israel, was closed and empty because of the rocket threat.
Israel has three main demands: an end to Palestinian attacks, international supervision of any truce, and a halt to Hamas rearming. Hamas demands an end to Israeli attacks and the opening of border crossings to vital cargo.
Livni said the operation was designed to change the rules of Israel's struggle against Hamas after years of firing rockets at Israel. From now on, she said, "when Israel is targeted, Israel is going to retaliate."
Israeli military spokeswoman Maj. Avital Leibovich said Hamas was to blame for civilian casualties because it operates in densely populated areas.
"If Hamas chose cynically to use those civilians as human shields, then Hamas should be accountable," she said.
Hamas leader Mahmoud Zahar exhorted Palestinians to fight the Israeli forces and target Israeli civilians and Jews abroad.
"The Zionists have legitimized the killing of their children by killing our children. They have legitimized the killing of their people all over the world by killing our people," Zahar said in a grainy video broadcast on Hamas TV.
Israel's operation has sparked anger across the Arab world and has drawn criticism from countries such as Turkey, Egypt and Jordan, which have ties with Israel and have been intimately involved in Mideast peacemaking.
In Beirut, Lebanon, protesters tried to pull away barbed wire blocking their path to the U.S. Embassy. They were driven back with heavy blasts of water.