Jack Osbourne Thanks Fans for 'Inspirational Words' After Revealing Diagnosis

"It means a lot," the new dad, who has multiple sclerosis, Tweeted before stepping out with his dog

Photo: Deano/Splash News Online

Jack Osbourne has the support of one of his biggest fans.

A day after her younger brother revealed he had been diagnosed with multiple sclerosis, Kelly Osbourne showed her respect and admiration for him on Twitter as she asked others to do the same.

“Please support my brother @mrjacko in not only his bravery but honesty!” his sister Tweeted. “i love you so much jack & I’m so proud of you!”

RELATED: Sharon Osbourne Brought to Tears over Jack Osbourne’s Illness

As for Osbourne, 26, who found out he had the disease (which affects the central nervous system) just two weeks after he and fiancée Lisa Stelly welcomed daughter Pearl Clementine in April, he had nothing but gratitude for all of those who expressed their support.

“Thank you all so much for the kind and inspirational words,” he Tweeted. “It means a lot. #adaptandovercome.”

The reality star seemed to keep with his routine, stepping out to walk his dog Monday near his home in Los Angeles.

For all the details of Jack’s shocking diagnosis, a candid interview with Jack and Sharon and more on how the famous family is coping, pick up this week’s issue of PEOPLE, on newsstands Friday