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We consider local weak solutions to the widely degenerate parabolic PDE ∂ t u - div (| D u | - λ) + p - 1 Du | D u | = f in Ω T = Ω × (0 , T) , where p ≥ 2 , Ω is a bounded domain in R n for n ≥ 2 , λ is a non-negative constant and · + stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Besov parabolic space when p > 2 and that f ∈ L loc 2 (Ω T) if p = 2 , we prove the Sobolev spatial regularity of a novel nonlinear function of the spatial gradient of the weak solutions. This result, in turn, implies the existence of the weak time derivative for the solutions of the evolutionary p-Poisson equation. The main novelty here is that f only has a Besov or Lebesgue spatial regularity, unlike the previous work [7], where f was assumed to possess a Sobolev spatial regularity of integer order. We emphasize that the results obtained here can be considered, on the one hand, as the parabolic analog of some elliptic results established in [6], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations. [ABSTRACT FROM AUTHOR]

Consider the well-posedness problem of an anisotropic parabolic equation with a nonstandard growth order. The weak solution is introduced, an L ∞ -estimate is provided, and the existence is established using the parabolically regularized method. However, the weak solution under consideration is not in L 1 (0 , T ; W 0 1 , p → (x) (Ω)) , and defining the trace becomes a new problem. In this paper, we give a new definition for the trace of u ∈ L ∞ (Q T) to solve this problem. Based on the new concept of the generalized trace and the selection of suitable test functions, the stability theorems of the weak solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form u t = a i ′ j ′ u i ′ j ′ + 2 x n γ / 2 a i ′ n u i ′ n + x n γ a nn u nn + b i ′ u i ′ + x n γ / 2 b n u n + c u + f (γ ≤ 1). When the equation above is singular, it can be derived from Monge–Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution u, which is called s-polynomial. To prove C s 2 + α -regularity and higher regularity of the solution u, we establish generalized Schauder theory which approximates coefficients of the operator with s-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity. [ABSTRACT FROM AUTHOR]

We consider the homogeneous Dirichlet problem for the parabolic equation ut−div(∣∇u∣p(x,t)−2∇u)=f(x,t)+F(x,t,u,∇u){u}_{t}-{\rm{div}}({| \nabla u| }^{p\left(x,t)-2}\nabla u)=f\left(x,t)+F\left(x,t,u,\nabla u) in the cylinder QT≔Ω×(0,T){Q}_{T}:= \Omega \times \left(0,T), where Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N}, N≥2N\ge 2, is a C2{C}^{2}-smooth or convex bounded domain. It is assumed that p∈C0,1(Q¯T)p\in {C}^{0,1}\left({\overline{Q}}_{T}) is a given function and that the nonlinear source F(x,t,s,ξ)F\left(x,t,s,\xi ) has a proper power growth with respect to ss and ξ\xi . It is shown that if p(x,t)>2(N+1)N+2p\left(x,t)\gt \frac{2\left(N+1)}{N+2}, f∈L2(QT)f\in {L}^{2}\left({Q}_{T}), ∣∇u0∣p(x,0)∈L1(Ω){| \nabla {u}_{0}| }^{p\left(x,0)}\in {L}^{1}\left(\Omega ), then the problem has a solution u∈C0([0,T];L2(Ω))u\in {C}^{0}\left(\left[0,T];\hspace{0.33em}{L}^{2}\left(\Omega )) with ∣∇u∣p(x,t)∈L∞(0,T;L1(Ω)){| \nabla u| }^{p\left(x,t)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega )), ut∈L2(QT){u}_{t}\in {L}^{2}\left({Q}_{T}), obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: ∣∇u∣2(p(x,t)−1)+r∈L1(QT),for any 0

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE u t - div ⁡ ( (| D ⁢ u | - ν) + p - 1 ⁢ D ⁢ u | D ⁢ u | ) = f in ⁢ Ω T = Ω × (0 , T) , where Ω is a bounded domain in ℝ n for n ≥ 2 , p ≥ 2 , ν is a positive constant and (⋅) + stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative u t . The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius ν centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations [corrected version of mr2584740], J. Math. Pures Appl. (9) 93 2010, 6, 652–671], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations. [ABSTRACT FROM AUTHOR]

We study the symmetric stochastic p-Stokes system, p ∈ (1 , ∞) , in a bounded domain. The results are two-fold: First, we show that in the context of analytically weak solutions, the stochastic pressure—related to non-divergence free stochastic forces—enjoys almost - 1 / 2 temporal derivatives on a Besov scale. Second, we verify that the velocity u of strong solutions obeys 1/2 temporal derivatives in an exponential Nikolskii space. Moreover, we prove that the non-linear symmetric gradient V (ϵ u) = (κ + ϵ u) (p - 2) / 2 ϵ u , κ ≥ 0 , which measures the ellipticity of the p-Stokes system, has 1/2 temporal derivatives in a Nikolskii space. [ABSTRACT FROM AUTHOR]

We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation ∂ t (| u | p - 2 u) + (- Δ p) s u = 0 for p ∈ (1 , ∞) and s ∈ (0 , 1) . Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even in the linear case of the nonlocal heat equation and in the time-independent case of fractional p - Laplace equation, our approach provides an alternate route to Harnack estimates without using Moser iteration, log estimates or Krylov–Safanov covering arguments. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following system with nonlinear signal consumption u t = Δ γ v u + r u - μ u α , x ∈ Ω , t > 0 , v t = Δ v - u β v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ∈ R n n ≥ 2 . It shown that whenever r > 0 , μ > 0 , α > 2 , β > 0 and α β > n + 2 2 , then the original system will produce a global classical solution and the solution converges to equilibrium r μ 1 α - 1 , 0 as t → ∞. [ABSTRACT FROM AUTHOR]

Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional p-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case. [ABSTRACT FROM AUTHOR]

We study the so-called expansion of positivity for doubly nonlinear parabolic type equations, including the porous medium and evolutionary p-Laplace equations. The expansion of positivity yields an alternative proof of the fundamental Hölder regularity for the evolutional p-Laplace equations, originally due to DiBenedetto. [ABSTRACT FROM AUTHOR]

We consider a class of degenerate equations in non-divergence form satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin diffusions. [ABSTRACT FROM AUTHOR]

We carry on the investigation started in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3] about the regularity of weak solutions to the strongly degenerate parabolic equation u t - div ⁡ [ (| D ⁢ u | - 1) + p - 1 ⁢ D ⁢ u | D ⁢ u | ] = f in ⁢ Ω T = Ω × (0 , T) , where Ω is a bounded domain in ℝ n for n ≥ 2 , p ≥ 2 and (⋅) + stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that f ∈ L loc p ′ ⁢ (0 , T ; B p ′ , ∞ , loc α ⁢ (Ω)) , with α ∈ (0 , 1) and p ′ = p / (p - 1) . This leads us to obtain higher fractional differentiability results for a function of the spatial gradient Du of the solutions. Moreover, we establish the higher summability of Du with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [P. Ambrosio, Besov regularity for a class of singular or degenerate elliptic equations, J. Math. Anal. Appl. 505 2022, 2, Paper No. 125636], and on the other hand as the fractional version of some results established in [P. Ambrosio and A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, preprint 2022, version 3, https://arxiv.org/abs/2204.05966v3]. [ABSTRACT FROM AUTHOR]

In this paper, we study two related features of the regularity of the weak solutions to the following strongly degenerate parabolic equation u t - div D u - 1 + p - 1 Du D u = f in Ω T = Ω × (0 , T) , where Ω is a bounded domain in R n for n ≥ 2 , p ≥ and T > 0 . We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that f ∈ L loc 2 Ω T . This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum f. [ABSTRACT FROM AUTHOR]

We define a homogeneous De Giorgi class of order p = 2 that contains the solutions of evolution equations of the types ξ ⁢ (x , t) ⁢ u t + A ⁢ u = 0 and (ξ ⁢ (x , t) ⁢ u) t + A ⁢ u = 0 , where ξ > 0 almost everywhere and A is a suitable elliptic operator. For functions belonging to this class, we prove a Harnack inequality. As a byproduct, one can obtain Hölder continuity for solutions of a subclass of the first equation. [ABSTRACT FROM AUTHOR]

We study a general form of a degenerate or singular parabolic equation u t - | D u | γ (Δ u + (p - 2) Δ ∞ N u) = 0 that generalizes both the standard parabolic p-Laplace equation and the normalized version that arises from stochastic game theory. We develop a systematic approach to study second order Sobolev regularity and show that D 2 u exists as a function and belongs to L loc 2 for a certain range of parameters. In this approach proving the estimate boils down to verifying that a certain coefficient matrix is positive definite. As a corollary we obtain, under suitable assumptions, that a viscosity solution has a Sobolev time derivative belonging to L loc 2 . [ABSTRACT FROM AUTHOR]

We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet problem for the parabolic equations ut−div(A(z,∣∇u∣)∇u)=F(z,u,∇u),z=(x,t)∈Ω×(0,T),{u}_{t}-{\rm{div}}\left({\mathcal{A}}\left(z,| \nabla u| )\nabla u)=F\left(z,u,\nabla u),\hspace{1.0em}z=\left(x,t)\in \Omega \times \left(0,T), with the double phase flux A(z,∣∇u∣)∇u=(∣∇u∣p(z)−2+a(z)∣∇u∣q(z)−2)∇u{\mathcal{A}}\left(z,| \nabla u| )\nabla u=(| \nabla u{| }^{p\left(z)-2}+a\left(z)| \nabla u{| }^{q\left(z)-2})\nabla u and the nonlinear source FF. The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions aa, pp, and qq are Lipschitz-continuous, a(z)a\left(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents pp, and qq satisfy the balance conditions 2NN+20\varepsilon \gt 0.

A bacterial biofilm is an aggregate of micro-organisms growing fixed onto a solid surface, rather than floating freely in a liquid. Biofilms play a major role in various practical situations such as surgical infections and water treatment. We consider a non-linear partial differential equation (PDE) model of biofilm growth subject to initial and Dirichlet boundary conditions, and the inverse coefficient problem of recovering the unknown parameters in the model from extra measurements of quantities related to the biofilm and substrate. By addressing and analysing this inverse problem, we provide reliable and robust reconstructions of the primary physical quantities of interest represented by the diffusion coefficients of substrate and biofilm, the biomass spreading parameters, the maximum specific consumption and growth rates, the biofilm decay rate and the half saturation constant. We give particular attention to the constant coefficients involved in the leading-part non-linearity, and present a uniqueness proof and some numerical results. In the course of the numerical investigation, we have identified extra data information that enables improving the reconstruction of the eight-parameter set of physical quantities associated to the model of biofilm growth. [ABSTRACT FROM AUTHOR]

We study the parabolic equations involving local and nonlocal p-Laplace-type operators. Under the assumption that 2 < p < ∞ , an intrinsic Harnack inequality of nonnegative weak solutions to such problem is established by employing the energy estimates, De Giorgi lemma, and measure theoretical argument. [ABSTRACT FROM AUTHOR]

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity ∂ t u = [ | D u | q + a (x , t) | D u | s ] Δ u + (p - 2) D 2 u Du | D u | , Du | D u | , where 1 < p < ∞ , - 1 < q ≤ s < ∞ and a (x , t) ≥ 0 . The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of q, s, such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when q = p - 2 and q < s , it will encompass the parabolic p-Laplacian both in divergence form and in non-divergence form. We aim to explore the L ∞ to C 1 , α regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local Hölder regularity of spatial gradients of viscosity solutions. [ABSTRACT FROM AUTHOR]

We show interior Hölder continuity for a class of quasi-linear degenerate reaction-diffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term is locally bounded except in zero. The class of equations we analyze is motivated by a model that describes the growth of biofilms. Our method is based on the original proof of interior Hölder continuity for the porous medium equation. We do not restrict ourselves to solutions that are limits in the weak topology of a sequence of approximate continuous solutions of regularized problems, which is a common assumption. [ABSTRACT FROM AUTHOR]

We show that signed weak solutions to parabolic obstacle problems with porous medium-type structure are locally Hölder continuous, provided that the obstacle is Hölder continuous. [ABSTRACT FROM AUTHOR]

Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the p-Laplacian type diffusion is also considered. [ABSTRACT FROM AUTHOR]

We study the free boundary of the porous medium equation with nonlocal drifts in dimension one. Under the assumption that the initial data has super-quadratic growth at the free boundary, we show that the solution is smooth in space and C loc 2 , 1 in time, and then the free boundary is C loc 2 , 1 . Moreover if the drift is local, both the solution and the free boundary are smooth. [ABSTRACT FROM AUTHOR]

In this paper, we study the second-order Sobolev regularity of solutions to the parabolic p-Laplace equation. For any p-parabolic function u, we show that D (D u p - 2 + s 2 D u) exists as a function and belongs to L loc 2 with s > - 1 and 1 < p < ∞ . The range of s is sharp. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish new pointwise regularity results for solutions to degenerate second-order partial differential equations with a Kolmogorov-type operator of the form L : = ∑ i , j = 1 m ∂ x i x j 2 + ∑ i , j = 1 N b ij x j ∂ x i - ∂ t , where (x , t) ∈ R N + 1 , 1 ≤ m ≤ N and the matrix B : = (b ij) i , j = 1 , … , N has real constant entries. In particular, we show that if the modulus of L p -mean oscillation of L u at the origin is Dini, then the origin is a Lebesgue point of continuity in L p average for the second-order derivatives ∂ x i x j 2 u , i , j = 1 , … , m , and the Lie derivative ∑ i , j = 1 N b ij x j ∂ x i - ∂ t u . Moreover, we are able to provide a Taylor-type expansion up to second order with an estimate of the rest in L p norm. The proof is based on decay estimates, which we achieve by contradiction, blow-up and compactness results. [ABSTRACT FROM AUTHOR]

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on Z d . In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment. [ABSTRACT FROM AUTHOR]

In this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form ∂ t u - div A (x , t , u , D u m) = div F , where the vector field A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents m > 0 and p > 1 with m (p - 1) > 1 are included in our considerations. [ABSTRACT FROM AUTHOR]

We introduce the notion of kinetic maximal L 2 -regularity with temporal weights for the (fractional) Kolmogorov equation. In particular, we determine the function spaces for the inhomogeneity and the initial value which characterize the regularity of solutions to the fractional Kolmogorov equation in terms of fractional anisotropic Sobolev spaces. It is shown that solutions of the homogeneous (fractional) Kolmogorov equation define a semi-flow in a suitable function space and the property of instantaneous regularization is investigated. [ABSTRACT FROM AUTHOR]

We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. ∂ t u - div (m | u | m - 1 D u) = 0 , m > 0. As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions. [ABSTRACT FROM AUTHOR]

The author presents a simplified proof for the local continuity of the weak solutions to the porous medium equation with variable positive bounded exponent γ (x , t) u t - ∇ · | u | γ (x , t) ∇ u = 0 , in Ω T = Ω × (0 , T ] , 0 < T < ∞. . [ABSTRACT FROM AUTHOR]

A regularity for a parabolic p-Laplacian system (p > 2) is studied by the use of the discrete Morse flow method which is known as one of the ways to approximate a solution to parabolic partial differential equations. Our approximate solution is constructed from the sequence of minimizers of variational functionals whose Euler–Lagrange equations are the time discretized p-Laplacian system. The aim of this paper is to establish that the regularity estimates for the approximate solution hold uniformly on two approximation parameters and show strong convergence of the approximate solution. [ABSTRACT FROM AUTHOR]

We obtain sharp parabolic interior and global Schauder estimates for solutions to nonlocal space-time master equations (∂ t + L) s u = f in R × Ω , where L is an elliptic operator in divergence form, subject to homogeneous Dirichlet and Neumann boundary conditions. In particular, we establish the precise behavior of solutions near the boundary. Along the way, we prove a characterization of the correct intermediate parabolic Hölder spaces in the spirit of Sergio Campanato. [ABSTRACT FROM AUTHOR]

In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

Using the sub-supersolution method, we study the existence of positive solutions for the anisotropic problem 0.1 - ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i - 2 ∂ u ∂ x i = λ u q - 1 where Ω is a bounded and regular domain of R N , q > 1 , and λ > 0 . [ABSTRACT FROM AUTHOR]

Given a second-order partial differential operator P = P o + X + a on an open set Ω in R n , where P o is the principal part and X is a real vector field, with non-negative real characteristic form, we study the s ′ -Gevrey regularity on an open subset ω of Ω , of s-Gevrey vectors of P on ω . For that we associate to any subset A ⊂ Ω an integer (finite or + ∞ ) named the type of A with respect to P o and denoted τ (A ; P o ) (see in the next sections, precise definitions, facts and remarks about it). Denoting the space of s-Gevrey vectors of P in ω by G s (ω , P) , we prove that G s (ω ; P) ⊂ G s ′ (ω) , with s ′ = τ (ω ; P) · s under the assumption that the coefficients of P are in G s (ω) . Moreover, s ′ is optimal. [ABSTRACT FROM AUTHOR]

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains. [ABSTRACT FROM AUTHOR]

We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given by∂tu−Δ(|u|m−1u)=divF,where m>1. More precisely, we prove that under suitable assumptions the spatial gradient D(|u|m−1u) of any weak solution is integrable to a larger power than the natural power 2. Our analysis includes both the case of the lateral boundary and the initial boundary. [ABSTRACT FROM AUTHOR]

We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation

In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type ( 2 ≤ p < ∞ ) under a strong absorption condition: Δ p u − ∂ u ∂ t = λ 0 u + q in Ω T : = Ω × ( 0 , T ) , where 0 ≤ q < 1 . This model is interesting because it yields the formation of dead-core sets, i.e., regions where non-negative solutions vanish identically. We shall prove sharp and improved parabolic C α regularity estimates along the set F 0 ( u , Ω T ) = ∂ { u > 0 } ∩ Ω T (the free boundary), where α = p p − 1 − q ≥ 1 + 1 p − 1 . Some weak geometric and measure theoretical properties as non-degeneracy, positive density, porosity and finite speed of propagation are proved. As an application, we prove a Liouville-type result for entire solutions. A specific analysis for Blow-up type solutions will be done as well. The results are new even for dead-core problems driven by the heat operator. [ABSTRACT FROM AUTHOR]

We study the continuity of solutions of the homogeneous Dirichlet problem for the equation u t − div | ∇ u | p ( x , t ) − 2 ∇ u = f in Q T = Ω × ( 0 , T ] , where Ω ⊂ R n , n ≥ 2 , is a domain with the boundary ∂ Ω ∈ C 2 + α and p ( x , t ) is a given function such that 2 n n + 2 < p − ≤ p ( x , t ) ≤ p + < ∞ , p ± = c o n s t . It is shown that if p , f are Hölder-continuous in Q ¯ T and | ∇ u 0 | p ( x , 0 ) ∈ L 1 ( Ω ) , then u ∈ C 1 , 1 2 ( Q ¯ T ∩ { t ≥ ϵ } ) for every ϵ > 0 . Moreover, if f is Lipschitz-continuous in t , p ≡ p ( x ) , ∇ p ∈ L ∞ ( Ω ) and either p ( x ) ≥ p − > 2 , or 2 n n + 2 < p − ≤ p ( x ) ≤ p + < 2 , then the solutions are Lipschitz-continuous in time in Q ¯ T ∩ { t ≥ ϵ } . [ABSTRACT FROM AUTHOR]

We study general parabolic equations of the form $$u_t = \text{ div }\,\mathbf {A}(x,t, u,D u) +\text{ div }\,(|\mathbf {F}|^{p-2} \mathbf {F})+ f$$ whose principal part depends on the solution itself. The vector field $$\mathbf {A}$$ is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderón-Zygmund estimates for locally bounded weak solutions to the equations when $$p>2n/(n+2)$$ . This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto (Degenerate parabolic equations, Springer, New York, 1993) and the maximal function free approach by Acerbi and Mingione (Duke Math J 136(2):285-320, 2007). [ABSTRACT FROM AUTHOR]

We define a homogeneous De Giorgi class of order p ⩾ 2 which suits two classes of evolution equations of the type μ ( x , t ) u t + A u = 0 and ( μ ( x , t ) u ) t + A u = 0 , where μ can be positive, null and negative and A a suitable elliptic operator. For functions belonging to this class we prove a local boundedness result. [ABSTRACT FROM AUTHOR]