How did the townspeople finally solve the problem with the smell in "A Rose for Emily"?

Though the townspeople talk a lot about the smell coming from Miss Emily's home, it's not the townspeople that "solve" the problem with the smell. The smell is solved by time...though the townspeople likely think they solved the problem by spreading lime around the grounds. 

The smell is first mentioned at the beginning of part two: 

So SHE vanquished them, horse and foot, just as she had vanquished their fathers thirty years before about the smell.

That was two years after her father's death and a short time after her sweetheart--the one we believed would marry her --had deserted her. 

This is an excellent example of foreshadowing; readers know at the end of the story that "her sweetheart" had not "deserted her" at all. In fact, the smell was coming from his decomposing body, which she had laid to rest in her bed. 

The townspeople are aware of the smell and discuss some possible options to deal with the problem. They blame it on the fact that a man, Miss Emily's house man, Tobe, can't "keep a kitchen properly." One woman asks a judge in town to deal with the problem.

"But what will you have me do about it, madam?" he said.

"Why, send her word to stop it," the woman said. "Isn't there a law? "

"I'm sure that won't be necessary," Judge Stevens said. "It's probably just a snake or a rat that nigger of hers killed in the yard. I'll speak to him about it."

The judge gets two more complaints:

"We really must do something about it, Judge. I'd be the last one in the world to bother Miss Emily, but we've got to do something." That night the Board of Aldermen met--three graybeards and one younger man, a member of the rising generation.

"It's simple enough," he said. "Send her word to have her place cleaned up. Give her a certain time to do it in, and if she don't. .."

"Dammit, sir," Judge Stevens said, "will you accuse a lady to her face of smelling bad?"

At this point, the judge orders several men to go to Miss Emily's house at night and sprinkle lime around the outside of the house and the grounds. They do all of this without confronting or discussing it with Miss Emily. She catches them in the act and they sneak away quickly. 

The problem appears to have been solved: "After a week or two the smell went away." Readers may be led to believe throughout the majority of the story that the townpeople actually did fix the problem with the lime and that it was caused by a rat or poor housekeeping, but, by the end of the story, readers know the truth.