Global estimates for solutions of singular parabolic and elliptic equations with variable nonlinearity

Strong solutions of evolution equations with p(x,t)-Laplacian: Existence, global higher integrability of the gradients and second-order regularity

“…In the present work, we are interested in the existence of strong solutions of problem (1.1) and their global regularity properties. Although these questions have already been addressed in a number of works, all known results refer to the singular equation (1.1) with 2N N +2 ≤ p(x, t) ≤ 2, or to the equations with p(x, t) nonincreasing in t. It is known [8,6,22] that the weak solution becomes a strong solution with u t ∈ L 2 (Q T ) and |∇u| p(x,t) ∈ L ∞ (0, T ; L 1 (Ω)), provided that |∇u 0 | p(x,0) ∈ L 1 (Ω), f ∈ L 2 (Q T ), p t ∈ L ∞ (Q T ) and either p t ≤ 0 a.e. in Q T , or |p t | ≤ C a.e.…”

“…x i x j u| p(x,t) ∈ L 1 (Q T ) [8,6], or D 2 x i x j u ∈ L 2 (Ω × (ǫ, T )) for every ǫ ∈ (0, T ) [6,5]. The strong solution may be Hölder or even Lipschitz continuous in t in the cylinders Ω × (ǫ, T ) with ǫ > 0, [22,24].…”

“…The strong solution may be Hölder or even Lipschitz continuous in t in the cylinders Ω × (ǫ, T ) with ǫ > 0, [22,24]. It is proven in [6] that if the initial function possesses a second-order regularity with respect to x and satisfies certain compatibility conditions, |f | p ′ (x,t) ∈ L 1 (Q T ) and f t ∈ L 2 (Q T ), then the singular equation with the Lipschitz-continuous exponent p(x, t) ≤ 2 in a convex C 2 domain has a unique strong solution such that…”

Parabolic Harnack Estimates for anisotropic slow diffusion

“…Finite Speed Propagation. In the context of non-Newtonian fluids, already in the 80s some techniques were known in order to estimate the support of solutions by comparison (see for instance [20], [56]) or by energy methods (see [2]). A further step in doubly nonlinear equations was made in [3], where the energy methods were fully exploited to avoid the comparison principle, which is not available in this case.…”

“…To prove (2) we note that the condition p1 ≥ α + 1 is stronger than pα+1 ≥ α + 1, so we may use the conclusion of (2). Take q ∈ [α+1, p1 ).…”