Global estimates for solutions of singular parabolic and elliptic equations with variable nonlinearity. (June 2020) - Antontsev, Stanislav; Shmarev, Sergey
Abstract: We consider the homogeneous Dirichlet problem for the equation (0.1) u t = div ( ϵ 2 + | ∇ u | 2 ) p ( x, t ) − 2 2 ∇ u + f ( x, t ), ϵ ≥ 0, in the cylinder Q T = Ω × ( 0, T ), Ω ⊂ R d, d ≥ 2, with the variable exponent 2 d d + 2 < p − ≤ p ( x, t ) ≤ p + ≤ 2, p ± = c o n s t . We find sufficient conditions on p, ∂ Ω, f and u ( x, 0 ) which provide the existence of solutions with the following global regularity properties: u t ∈ L ∞ ( 0, T ; L 2 ( Ω ) ), | ∇ u | ∈ C 0 ( [ 0, T ] ; L 2 ( Ω ) ), | u t | p p − 1, | u x i x j | p, | ∇ u t | p, ( ϵ 2 + | ∇ u | 2 ) p − 2 2 | u x i x j | 2 ∈ L 1 ( Q T ), p = p ( x, t ), i, j = 1, 2, …, d . For the solutions of the stationary counterpart of Eq. (0.1), div ( ϵ 2 + | ∇ v | 2 ) p 0 ( x ) − 2 2 ∇ v = Φ ( x ) in Ω, v = 0 on ∂ Ω, the inclusions | v x i x j | p 0, ( ϵ 2 + | ∇ v | 2 ) p 0 − 2 2 | v x i x j | 2 ∈ L 1 ( Ω ) are established.