Until last month, the neighbors never saw much of the family living in the rundown house on First Avenue in Massapequa Park on Long Island.

But in the five weeks since the authorities charged the house’s owner, Rex Heuermann, in the Gilgo Beach serial killings, his wife and children have become unlikely fixtures in their neighborhood.

The family — Mr. Heuermann’s wife, Asa Ellerup, 59, and their children, Victoria, 26, and Christopher, 33 — slipped out of the house in July just before crowds of reporters and gawkers descended and investigators began to hunt for evidence in a search that lasted nearly two weeks.

But Ms. Ellerup and the children soon returned and quickly became a daily presence outside the house, sitting together on the front porch or working to put the place back together. She declined to speak to a reporter who recently stopped by.

Not so long ago, the family had a reputation as reclusive. Now, while they still have little contact with the neighbors, clouds of savory smoke regularly waft from their yard.

“They’re having barbecues on the front lawn — they never did anything like that before,” said Etienne de Villiers, 68, a retired New York City firefighter who lives next door. “Suddenly, they’re out there all the time.”

A lawyer representing the two children offered an explanation for the change in behavior: Investigators had left the house uninhabitable.

“It’s literally piled floor-to-ceiling with debris,” the lawyer, Vess Mitev, said. “It’s like someone broke in and tore the place apart.”

The unsolved Gilgo Beach case riveted the public for more than a decade, and Mr. Heuermann’s arrest on July 13 was a huge break in it. Now, a bizarre public battle is unfolding over the investigators’ search of his ramshackle home, with his family recently holding a news conference on the lawn.

The family always stood out as aloof in the otherwise tight-knit neighborhood, their dilapidated ranch house drawing sneers on a mostly fastidiously tidy block. Many neighbors presumed Ms. Ellerup and the children would never return to the home given its new notoriety.

“Who in their right mind would come back to a house equated with the most horrible thing to ever happen in this town?” one neighbor, Chris Duncan said.

Mr. Heuermann, who is being held without bail at a Suffolk County jail, is scheduled to appear in court next month on charges of murdering three of the 11 victims whose bodies were found more than a decade ago along Ocean Parkway, near Gilgo Beach. He has pleaded not guilty.

Prosecutors say he hired the women as escorts and then killed them, bound their bodies and wrapped them in burlap. He is the prime suspect in the death of a fourth woman. It is unclear whether he is linked to the other seven victims.

To many homeowners in this suburban village, Mr. Heuermann was a workaday, if antisocial, neighbor who commuted to and from his architectural consulting firm in Manhattan. Last year, though, after repeatedly running into dead ends, investigators used DNA analysis and cellphone records to close in on him.

The family’s lawyers are seeking to address the damage they say investigators did to the home in the search for evidence. The lawyers have associates helping Ms. Ellerup and the children around the house and recently set up a lectern on the front lawn to publicly air their complaints.

John Ray, a lawyer who represents the families of two women whose remains were found in the Gilgo Beach area but whom Mr. Heuermann has not been charged with killing, held his own news conference and called for Ms. Ellerup to be investigated.

Raymond Tierney, the Suffolk County district attorney, has said Mr. Heuermann, 59, committed the killings while his family was away on trips.

Mr. Ray said it was implausible that Ms. Ellerup was ignorant of her husband’s deeds and “should be considered a suspect and investigated accordingly.”

A lawyer for Ms. Ellerup, Robert A. Macedonio, dismissed Mr. Ray’s assertion.

“She knew nothing about any of this,” Mr. Macedonio said in a phone interview. “If it happened, he was leading a complete double life.”

Mr. Macedonio said that investigators had never interviewed Ms. Ellerup or the couple’s children and that there was no indication they would be charged.

“She hasn’t even begun to process what he’s being charged with,” Mr. Macedonio said. “She’s still putting her own life back together.”

Ms. Ellerup has not visited Mr. Heuermann in jail but did speak to him once by phone when he called, Mr. Macedonio said.

Upon returning to the family’s house, she greeted the media scrum outside with a lewd gesture and settled into bantering from afar with reporters and photographers, by turns cordial and dismissive.

“They’re sitting outside smiling,” Mr. Ray said. “She seems very out of sync with what is really happening.”

Ms. Ellerup has filed for divorce “to protect herself and her children” and is considering suing the authorities for what she says is the extensive damage investigators caused, Mr. Macedonio said.

The family stayed with relatives and friends and slept in their car while the search proceeded, Mr. Macedonio said.

In terms of the damage to the house, the family’s lawyers say, investigators ripped up floorboards and pulled belongings out of closets, leaving them piled in disarray. The bathtub was cut open and mattresses were seized as evidence. Even now, Mr. Macedonio said, the family was sleeping on mats on the floor while waiting for new mattresses.

With their bedrooms in shambles, the children had to sleep in the basement, Mr. Mitev said.

Neighbors have not flocked to support the family. Many are unsure whether Ms. Ellerup deserves sympathy or suspicion, and are skeptical that she could have been clueless about the crimes her husband is accused of.

“We’re getting a slew of emails from people wanting to help and a slew from other people saying, ‘I hope you burn in hell,’” Mr. Mitev said.

“Some are sympathetic and some say, ‘When can we get this thing out of here?’” he added, regarding the unsightly home. “But there’s no getting it out of here because they have nowhere to go.”

For now, Mr. Mitev added: “They are going to rebuild; they want to fix it up,” He said investigators had violated the family’s rights with what he called their reckless search.

Adding to the family’s woes, Mr. Macedonio said, Ms. Ellerup was being treated for breast and skin cancer and could not afford the premium payments for the health insurance she had through Mr. Heuermann’s business.

With the family facing financial hardship, damage to their home and shock from the criminal charges, things could change, he said.

“At this point,” he said, “there’s no plan.”

Some help has come from afar. More than $50,000 has been raised for the family through an online fund-raiser created by Melissa Moore, the daughter of the serial killer Keith Hunter Jesperson, known as the “Happy Face Killer.”

Mr. Heuermann grew up with his parents and four siblings in the house, which is on a tight grid of streets an hour from Midtown Manhattan. He bought it from his family in the 1990s but let it fall into disrepair as neighbors renovated theirs and watched the property values soar.

Many neighbors feared the house would become a notorious landmark. Village authorities have installed street signs forbidding parking or even stopping nearby. The return of the family has not helped in that regard, another neighbor, Warren Ferchaw, said.

“I really think the best thing for everyone is for them to move on and for the house to be torn down once and for all,” Mr. Ferchaw said while walking his dog near the house on a recent evening as the family barbecued on the front lawn.

This does not seem imminent. Nearby Massapequa is Ms. Ellerup’s hometown. She grew up there after immigrating to the United States from Iceland with her family as a child, and has lived in her current home for the 27-year duration of her marriage to Mr. Heuermann.

“This is all that they’ve known, so there’s no real options,” Mr. Mitev said.

Last week, the family seemed to be settling back in, moving items between the house and the garage. Ms. Ellerup bristled when a reporter approached.

“Shoo, shoo,” she said, extending her hands as if dismissing a dog. “We’ve got too much work to do.”