Global higher regularity of solutions to singular p(x,t)-parabolic equations

Abstract We study the homogeneous Dirichlet problem for the equation u t = div ( | ∇ u | p ( x , t ) − 2 ∇ u ) + f ( x , t , u ) in the cylinder Q T = Ω × ( 0 , T ) , Ω ⊂ R d , d ≥ 2 . It is assumed that p ( x , t ) ∈ ( 2 d d + 2 , 2 ) and | ∇ p | , | p t | are bounded a.e. in Q T . We find conditions on p ( x , t ) , f ( x , t , u ) and u ( x , 0 ) sufficient for the existence of strong solutions, local or global in time. It is proven that the strong solutions possess the property of global higher regularity: u t ∈ L 2 ( Q T ) , | ∇ u | ∈ L ∞ ( 0 , T ; L 2 ( Ω ) ) , | D i j 2 u | p ( x , t ) ∈ L 1 ( Q T ) .