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For the non-stationary Stokes system, it is well-known that one can improve spatial regularity in the interior, but not near the boundary if it is coupled with the no-slip boundary condition. In this note we show that, to the contrary, spatial regularity can be improved near a flat boundary if it is coupled with the Navier boundary condition, with either infinite or finite slip length. The case with finite slip length is more difficult than the case with infinite slip length. [ABSTRACT FROM AUTHOR]

In this paper, we study the first Dirichlet eigenfunction of the weighted $ p $-Laplacian on a bounded domain in a complete weighted Riemannian manifold. By constructing gradient estimates for the first eigenfunction, we obtain some relationships among the first eigenvalues of weighted $ p $-Laplacian. [ABSTRACT FROM AUTHOR]

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p-Laplace equation --Δpu = μ with p ∈ (1, 2). The cases when p ∈ (2 -- 1/n, 2) and p ∈ (3n--2/2n--1, 2 -- 1/n] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379-418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p ∈ (1, 3n--2/2n--1)]. Interior and global modulus of continuity estimates of the gradients of solutions are also established. [ABSTRACT FROM AUTHOR]

In this paper, we study the gradient estimate for the positive solutions of the equation Δu + au(logu)p + bu = f on an almost Ricci soliton (Mn, g, X, λ). In a special case, when X = ∇h for a smooth function h, we derive a gradient estimate for an almost gradient Ricci soliton. [ABSTRACT FROM AUTHOR]

We prove an existence result for a p-Laplacian problem set in the whole Euclidean space and exhibiting a critical term perturbed by a singular, convective reaction. The approach used combines variational methods, truncation techniques, and concentration compactness arguments, together with set-valued analysis and fixed point theory. De Giorgi’s technique, a priori gradient estimates, and nonlinear regularity theory are employed to obtain local C1,α{C}^{1,\alpha } regularity of solutions, as well as their pointwise decay at infinity. The result is new even in the non-singular case, also for the Laplacian.

In this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions. [ABSTRACT FROM AUTHOR]

Let Ω ∩ ℝn be a convex polytope (n ≤ 3). The Ritz projection is the best approximation, in the W1,2 0 -norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, the gradient at any point in Ω is controlled by the Hardy--Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of interest in the analysis of PDEs immediately follows. Among those are weighted spaces, Orlicz spaces, and Lorentz spaces. [ABSTRACT FROM AUTHOR]

We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of p-Laplace type when 2 n n + 2 < p ≤ 2 . The result is based on a reverse Hölder inequality in intrinsic cylinders combining p-intrinsic and (p, q)-intrinsic geometries. A singular scaling deficits affects the range of q. [ABSTRACT FROM AUTHOR]

Abstract We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold ( M , g ) $(M, g)$ Δ u + a u ( ln u ) p + b u ln u = 0 , $$ \Delta u +au(\ln{u})^{p}+bu\ln{u}=0, $$ where a ≠ 0 $a\ne 0$ , b are two constants and p = k 1 2 k 2 + 1 ≥ 2 $p=\frac{k_{1}}{2k_{2}+1}\ge 2$ , here k 1 $k_{1}$ and k 2 $k_{2}$ are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution.

The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three. [ABSTRACT FROM AUTHOR]

Let (λ k) be a strictly increasing sequence of positive numbers such that ∑ k = 1 ∞ 1 λ k < ∞ . Let f be a bounded smooth function and denote by u = u f the bounded classical solution to u (x) - 1 2 ∑ k = 1 m D kk 2 u (x) + ∑ k = 1 m λ k x k D k u (x) = f (x) , x ∈ R m. It is known that the following dimension-free estimate holds: ∫ R m ∑ k = 1 m λ k (D k u (y)) 2 p / 2 μ m (d y) ≤ (c p) p ∫ R m | f (y) | p μ m (d y) , 1 < p < ∞ where μ m is the "diagonal" Gaussian measure determined by λ 1 , ... , λ m and c p > 0 is independent of f and m. This is a consequence of generalized Meyer's inequalities [4]. We show that, if λ k ∼ k 2 , then such estimate does not hold when p = ∞ . Indeed we prove sup f ∈ C b 2 (R m) , ‖ f ‖ ∞ ≤ 1 ∑ k = 1 m λ k (D k u f (0)) 2 → ∞ as m → ∞. This is in contrast to the case of λ k = λ > 0 , k ≥ 1 , where a dimension-free bound holds for p = ∞ . [ABSTRACT FROM AUTHOR]

We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold (M , g) Δ u + a u (ln u) p + b u ln u = 0 , where a ≠ 0 , b are two constants and p = k 1 2 k 2 + 1 ≥ 2 , here k 1 and k 2 are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the influence of the initial data and the forcing terms on the regularity of solutions to a class of evolution equations including linear and semilinear parabolic equations as the model cases, together with the nonlinear p-Laplacian equation. We focus our study on the regularity (in terms of belonging to appropriate Lebesgue spaces) of the gradient of the solutions. We prove that there are cases where the regularity of the solutions as soon as  t > 0  is not influenced at all by the initial data. We also derive estimates for the gradient of these solutions that are independent of the initial data and reveal, once again, that for this class of evolution problems, the real "actors of the regularity" are the forcing terms. [ABSTRACT FROM AUTHOR]

In this paper, we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in R n + 1 with speed ψ σ k α (λ) , where α > 1 k is a constant, σ k (λ) is the k-th elementary symmetric polynomial of the principal radii of curvature and ψ is a preassigned positive smooth function defined on S n . We prove that under some assumptions of ψ , the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known L p Christoffel-Minkowski problem u 1 - p (x) σ k (∇ 2 u + u I) = c ψ (x) for 1 < p < k + 1 . [ABSTRACT FROM AUTHOR]

We prove the uniform gradient estimates of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The large-time estimates in terms of initial data mass and nonlinearity structure were obtained. In the particular case, when the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound. [ABSTRACT FROM AUTHOR]

Supervised learning in deep neural networks is commonly performed using error backpropagation. However, the sequential propagation of errors during the backward pass limits its scalability and applicability to low-powered neuromorphic hardware. Therefore, there is growing interest in finding local alternatives to backpropagation. Recently proposed methods based on forward-mode automatic differentiation suffer from high variance in large deep neural networks, which affects convergence. In this paper, we propose the Forward Direct Feedback Alignment algorithm that combines Activity-Perturbed Forward Gradients with Direct Feedback Alignment and momentum. We provide both theoretical proofs and empirical evidence that our proposed method achieves lower variance than forward gradient techniques. In this way, our approach enables faster convergence and better performance when compared to other local alternatives to backpropagation and opens a new perspective for the development of online learning algorithms compatible with neuromorphic systems. • Forward Gradient methods suffer from high variance that hinders convergence. • Activity-Perturbed Forward Gradients can be used to learn derivatives as direct feedback connections. • Feedback learning acts as a momentum that reduces the gradient variance closer to backpropagation. • Feedback learning reduces the biasness of Direct Feedback Alignment. • Feedback learning enables learning in convolutional layers. [ABSTRACT FROM AUTHOR]

Let (N, g) be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations Δ u (x) + a (x) u (x) log u (x) + b (x) u (x) = 0 on N where a(x) is C2-smooth while b(x) is C1 and its parabolic counterparts (Δ − ∂ ∂ t ) u (x , t) + a (x , t) u (x , t) log u (x , t) + b (x , t) u (x , t) = 0 on N × [0, ∞) where a(x, t) and b(x, t) are C2 with respect to x ∊ N while are C1 with respect to the time t. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates. [ABSTRACT FROM AUTHOR]

In this paper, we establish the gradient and Pogorelov estimates for $ k $-convex-monotone solutions to parabolic $ k $-Hessian equations of the form $ -u_t\sigma_k(\lambda(D^2u)) = \psi(x, t, u) $. We also apply such estimates to obtain a Liouville type result, which states that any $ k $-convex-monotone and $ C^{4, 2} $ solution $ u $ to $ -u_t\sigma_k(\lambda(D^2u)) = 1 $ in $ \mathbb{R}^n\times(-\infty, 0] $ must be a linear function of $ t $ plus a quadratic polynomial of $ x $, under some growth assumptions on $ u $. [ABSTRACT FROM AUTHOR]

This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.

In this article, we present new gradient estimates for positive solutions to a class of non‐linear elliptic equations involving the f‐Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville‐type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed. [ABSTRACT FROM AUTHOR]

We show convergence of the gradients of the Schrödinger potentials to the (uniquely determined) gradient of Kantorovich potentials in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science. [ABSTRACT FROM AUTHOR]

This article studies a nonlinear parabolic equation on a complete weighted manifold where the metric and potential evolve under a super Perelman-Ricci flow. It derives elliptic gradient estimates of local and global types for the positive solutions and exploits some of their implications notably to a general Liouville type theorem, parabolic Harnack inequalities and classes of Hamilton type dimension-free gradient estimates. Some examples and special cases are discussed for illustration. [ABSTRACT FROM AUTHOR]

We are concerned with global gradient estimates of very weak solutions for a general class of quasilinear elliptic equations with nonstandard growth. We establish such estimates under basic structure assumptions on the nonlinearity and natural geometric assumption on the complement of the bounded domain. Our approach is based on a generalized Lipschitz truncation alongside a generalized self-improving property of thick sets. This work provides a careful analysis of the thickness described in Adimurthi and Phuc (Calc Var Partial Differ Equ 54:3107–3139, 2015), Harjulehto and Hästö (Z Anal Anwend 38(1):73–96, 2019), Heinonen et al (Potential theory of degenerate elliptic equations, Oxford University Press, Oxford, 1993). [ABSTRACT FROM AUTHOR]

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a (k , m) -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results. [ABSTRACT FROM AUTHOR]

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has ‖ P t μ − P t ν ‖ 1 ≤ W 1 (μ , ν) t for all probability measures μ , ν where P t is the heat semigroup and W 1 is the ℓ 1 -Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the eigenvalue estimate λ 1 ≥ log ⁡ (2) / diam 2. [ABSTRACT FROM AUTHOR]

In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with non-negative Gaussian curvature, the rigidity of the sharp Li-Yau gradient estimate for positive solutions to heat equations and the related estimates for Dirichlet Green’s functions on Riemannian manifolds with non-negative Ricci curvature. Moreover, we also obtain the corresponding stability results. [ABSTRACT FROM AUTHOR]

In this paper, we prove up to the boundary gradient estimates for viscosity solutions to inhomogeneous nonlinear Free Boundary Problems (FBP) governed by fully nonlinear and quasilinear elliptic equations with unbounded measurable ingredients. Here, we build upon our previous results in [9] to construct Inhomogeneous Pucci Barriers (IPB) for the Pucci extremal equations with unbounded coefficients. Using these barriers, we obtain a version of a boundary growth type lemma for inhomogeneous nonlinear equations that may be of independent interest. In a certain way, this lemma detects the expansion of the level sets of supersolutions from the boundary to the interior. The use of this boundary growth type lemma together with the geometry of IPB bridge the interchanging information between the free boundary condition and Dirichlet boundary data on the free and fixed boundary respectively. This produces an estimate on the trace of solutions to FBP along the fixed boundary. This way, control of such solutions (up to the boundary) by the distance to the negative phase is obtained. Finally, this distance control combined with the PDE boundary gradient estimates render our final result. [ABSTRACT FROM AUTHOR]

In this paper we establish a higher integrability result up to the boundary of weak solutions to doubly nonlinear parabolic systems. We show that the spatial gradient of a weak solution with vanishing lateral boundary values is integrable to a larger power than the natural power p, where the statement holds for parameters in the subquadratic case max { 2 n n + 2 , 1 } < p ≤ 2 . [ABSTRACT FROM AUTHOR]

In this paper, the following d -dimensional distribution-dependent stochastic differential equation driven by a pure jump process is studied: X t ξ = ξ + ∫ 0 t b (X s − ξ , μ s) d s + ∫ 0 t ∫ B 0 σ (X s − ξ , z , μ s) N ̂ (d z , d s) , t ∈ [ 0 , 1 ] , where μ s denotes the distribution of X s ξ . The differentiability of the map ξ ↦ X t ξ is investigated in the sense of L 2 (Ω ; ℝ d). By the Malliavin calculus for jump processes, the following Bismut type derivative formula is established, ∇ η f (X t ξ) = (f (X t ξ) M t ξ) , where f is a test function and M t ξ is a random variable depending on the initial value ξ. Sharp gradient estimates in short time are also obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the global boundedness and stability of classical solutions to an important alarm-taxis ecosystem that is significant in understanding the behaviors of prey and predators. Specifically, it studies the case where prey attracts the secondary predators when threatened by the primary predators. The secondary consumers pursue the signal generated by the interaction between the prey and direct consumers. However, obtaining the necessary gradient estimates for global existence seems difficult in the critical case due to the strong coupled structure. Therefore, a new approach is developed to estimate the gradient of prey and primary predators, which takes advantage of slightly higher damping power. Subsequently, the boundedness of classical solutions in two-dimension with Neumann boundary conditions can be established by energy estimates and semigroup theory. Moreover, by constructing Lyapunov functional, it is proved that the coexistence homogeneous steady states are asymptotically stable, and the convergence rate is exponential under certain assumptions on the system coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we consider gradient estimates for positive solutions to the following equation △V u + aup log u = 0 on complete noncompact Riemannian manifold with k -dimensional Bakry-Émery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of |∇u| with respect to the lower bound of the Bakry-Émery Ricci curvature. [ABSTRACT FROM AUTHOR]

We study the global regularity in weighted Lorentz spaces for the gradient of weak solutions to p-Laplace equations in an open-bounded non-smooth domain which satisfies a p-capacity condition. Moreover, the gradient estimate is presented in terms of fractional maximal operators which are related to the Riesz potential and the fractional derivative. Our main concern is to extend the results of Tran and Nguyen [New gradient estimates for solutions to quasi-linear divergence form elliptic equations with general Dirichlet boundary data. J Differ Equ. 2020;268(4):1427–1462] but in the weighted Lorentz spaces associated to the Muckenhoupt weights. The key technique that we use in this paper is the good-λ inequality and the interesting properties of the cut-off fractional maximal operators. [ABSTRACT FROM AUTHOR]

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Laplacians (− Δ) Ω s , with Ω ⊂ R N. Recall that the regional fractional Laplacians are generated by Lévy-type processes which are not allowed to jump outside Ω. We consider weak solutions to the equation (− Δ) Ω s w (x) = p. v. ∫ Ω w (x) − w (y) | x − y | N + 2 s d y = f (x) , for s ∈ (0 , 1) and Ω ⊂ R N , subject to zero Neumann or Dirichlet boundary conditions. The boundary conditions are defined by considering w as well as the test functions in the fractional Sobolev spaces H s (Ω) or H 0 s (Ω) respectively. While the interior regularity is well understood for these problems, little is known in the boundary regularity, mainly for the Neumann problem. Under optimal regularity assumptions on Ω and provided f ∈ L p (Ω) , we show that w ∈ C 2 s − N / p (Ω ‾) in the case of zero Neumann boundary conditions. As a consequence for 2 s − N / p > 1 , w ∈ C 1 , 2 s − N p − 1 (Ω ‾). As what concerned the Dirichlet problem, we obtain w / δ 2 s − 1 ∈ C 1 − N / p (Ω ‾) , provided p > N and s ∈ (1 / 2 , 1) , where δ (x) = dist (x , ∂ Ω). To prove these results, we first classify all solutions having a certain growth at infinity when Ω is a half-space and the right hand side is zero. We then carry over a fine blow up and some compactness arguments to get the results. [ABSTRACT FROM AUTHOR]

Solutions (u , v) to the chemotaxis system u t = ∇ ⋅ ((u + 1) m − 1 ∇ u − u (u + 1) q − 1 ∇ v) , τ v t = Δ v − v + u in a ball Ω ⊂ R n , n ≥ 2 , wherein m , q ∈ R and τ ∈ { 0 , 1 } are given parameters with m−q>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in L p (Ω) for some p > p 0 := n 2 (1 − (m − q)). For radially symmetric solutions, we show that, if u is only bounded in L p 0 (Ω) and the technical condition m > n − 2 p 0 n is fulfilled, then, for any α > n p 0 , there is C>0 with u (x , t) ≤ C | x | − α f o r a l l x ∈ Ω a n d t ∈ (0 , T max) , T max ∈ (0 , ∞ ] denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any α < n p 0 , then u would already be bounded in L p (Ω) for some p > p 0 . [ABSTRACT FROM AUTHOR]

We investigate a general class of the so-called double phase problems on bounded domains. We establish an optimal global Calderón-Zygmund theory for such a non-uniformly elliptic problem under the assumptions that the coefficients have a small BMO semi-norm and the domain is sufficiently flat. Our regularity results not only cover discontinuous coefficients but also non-smooth domains going beyond the Lipschitz category. [ABSTRACT FROM AUTHOR]

We investigate the boundary regularity of minimizers of convex integral functionals with nonstandard p , q -growth and with Uhlenbeck structure. We consider arbitrary convex domains Ω and homogeneous Dirichlet data on some part Γ ⊂ ∂ Ω of the boundary. For the integrand we assume only a non-standard p , q -growth condition. We establish Lipschitz regularity of minimizers up to Γ, provided the gap between the growth exponents p and q is not too large, more precisely if 1 < p ≤ q < p (1 + 2 n). To our knowledge, this is the first boundary regularity result under a non-standard p , q -growth condition. [ABSTRACT FROM AUTHOR]

This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.

Let be a nonnegative Schrödinger operator on , where and is a radially symmetric inverse square potential. Let be the operator norm of from the Lorentz space to , where. We establish both of upper and lower decay estimates of and study sharp decay estimates of. Furthermore, we characterize the Laplace operator from the view point of the decay of. [ABSTRACT FROM AUTHOR]

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on (m, 0) forms, i.e., sections of the canonical bundle of Kähler manifolds, where m is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of (m, 0) forms, which involves only the Ricci curvature and the gradient of the scalar curvature. [ABSTRACT FROM AUTHOR]

We discuss local gradient estimates of Li and Yau type on the smooth bounded positive solutions w : M × [ 0 , ∞) → R to a nonlinear evolution equation w t = Δ w + a (w - w 3) , where a > 0 is a constant, on a complete Riemannian manifold M . Global estimates are obtained from the local ones, the consequence of which will eventually yield classical Harnack inequalities for Parabolic Allen–Cahn equation and a Liouville type result for steady state solutions under the hypothesis of nonnegative Ricci curvature tensor. [ABSTRACT FROM AUTHOR]

Let (M , g , e - ϕ d ν) be a complete smooth metric measure space with the m-Bakry–Émery Ricci curvature bounded from below. In the first part of this article, we establish a local Hamilton-type gradient estimate for positive solutions to a nonlinear parabolic equation (Δ ϕ - q - ∂ t) u = a u (ln u) α on (M , g , e - ϕ d ν) , where q(x, t) is a smooth space-time function and a, α are real constants. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. In the second part of this article, we obtain a refined global gradient estimates for the positive solutions to the above nonlinear parabolic equation with a smooth function q(x) and α = 1 . As its application, we derive a uniform bound for the solutions to the corresponding elliptic equation. Our results generalize some recent works on this direction. [ABSTRACT FROM AUTHOR]