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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]

In this paper, we establish the boundary regularity results for viscosity solutions of fully nonlinear degenerate/singular parabolic equations of the form u t - x n γ F (D 2 u , x , t) = f , where γ < 1 . These equations are motivated by the porous media type or fast diffusion type equations. We show the boundary C 1 , α -regularity of functions in their solutions class and the boundary C 2 , α -regularity of viscosity solutions. As an application, we derive the global regularity results and the solvability of the Cauchy–Dirichlet problems. [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]

This paper deals with a class of fractional 1-Laplacian diffusion equations with variable orders, proposed as a model for removing multiplicative noise in images. The well-posedness of weak solutions to the proposed model is proved. To overcome the essential difficulties encountered in the approximation process, we place particular emphasis on studying the density properties of the variable-order fractional Sobolev spaces. Numerical experiments demonstrate that our model exhibits favorable performance across the entire image. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following quasilinear chemotaxis–consumption system with degenerate signal-dependent motility u t = ∇ · (v α ∇ u m) + α ∇ · (v α - 1 u ∇ v) + r u - μ u 2 , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under the homogeneous Neumann boundary condition in Ω ⊂ R n (n ≥ 1) with α , r , μ > 0 and m > 1 . It is shown that above system admits at least one global weak solution fulfilling the following boundedness property ‖ u (· , t) ‖ L p (Ω) + ‖ v (· , t) ‖ W 1 , ∞ (Ω) ≤ C for all t > 0 , p > 2 and m > 1 . This result gives an evident of the regularized effect of porous-medium-type diffusion. [ABSTRACT FROM AUTHOR]

We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase–plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. Finally, numerical simulations are presented that affirm the theoretical predictions on the existence, stability, and parametric dependence of the travelling waves. [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]

This work considers the doubly degenerate nutrient model u t = ∇ · u m - 1 v ∇ u - ∇ · f (u) v ∇ v + ℓ u v , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under no-flux boundary conditions in a smoothly bounded convex domain Ω ⊂ R n ( n ≤ 2 ), where the nonnegative function f ∈ C 1 ([ 0 , ∞)) is assumed to satisfy f (s) ≤ C f s α with α > 0 and C f > 0 for all s ≥ 1 . When m = 2 , it was shown that a global weak solution exists, either in one-dimensional setting with α = 2 , or in two-dimensional version with α ∈ 1 , 3 2 . The main results in this paper assert the global existence of weak solutions for 1 ≤ m < 3 and classical solutions for 3 ≤ m < 4 to the above system under the assumption α ∈ m - 1 , min m , m 2 + 1 if n = 1 , and m - 1 , min m , m 2 + 1 if n = 2 , which extend the range α ∈ (1 , 3 2) to α ∈ (1 , 2) in two dimensions for the case m = 2 . Our proof will be based on a new observation on the coupled energy-type functional and on an inequality with general form. [ABSTRACT FROM AUTHOR]

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic W 2 , p -regularity is imposed as an assumption. Also, we give a finite time blow-up criterion. [ABSTRACT FROM AUTHOR]

In this article, we study the inviscid limit of the solution to the Cauchy problem of a one-dimensional viscous conservation law, where the second-order term is nonlinear. Under the assumption that the inviscid equation admits a piecewise smooth solution with two noninteracting entropy shocks, we prove that the solution of the viscous equation converges uniformly to the piecewise smooth inviscid solution away from the shocks, even the strength of shocks is not small.

In this article, we study the mathematical properties of the solution u=(u1,…,uk){\bf{u}}=({u}^{1},\ldots ,{u}^{k}) to the degenerate parabolic system ut=∇⋅(∣∇u∣p−2∇u),(p>2).{{\bf{u}}}_{t}=\nabla \hspace{0.25em}\cdot \hspace{0.25em}({| \nabla {\bf{u}}| }^{p-2}\nabla {\bf{u}}),\hspace{1.0em}(p\gt 2). More precisely, we show the existence and uniqueness of solution u{\bf{u}} and investigate a priori L∞{L}^{\infty } boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component ul{u}^{l}, (1≤l≤k)(1\le l\le k), converges to the function clℬ{c}^{l}{\mathcal{ {\mathcal B} }} in space as t→∞t\to \infty . Here, the function ℬ{\mathcal{ {\mathcal B} }} is the fundamental or Barenblatt solution of pp-Laplacian equation, and the constant cl{c}^{l} is determined by the L1{L}^{1}-mass of ul{u}^{l}. The proof is based on the existence of entropy functional. As an application of the asymptotic large-time behaviour, we establish a Harnack-type inequality, which makes the size of the spatial average controlled by the value of the solution at one point.

This paper is concerned with an initial-boundary value problem of full compressible magnetohydrodynamics (MHD) equations on 3D bounded domains subject to non-slip boundary condition for velocity, perfectly conducting boundary condition for magnetic field, and homogeneous Dirichlet boundary condition for temperature. The global well-posedness of strong solutions with initial vacuum is established and the exponential decay estimates of the solutions are obtained, provided the initial total energy is suitably small. More interestingly, it is shown that for p ∈ (3 , 6) , the L p -norm of the gradient of density remains uniformly bounded for all t ≥ 0 . This is in sharp contrast to that in (Chen et al. in Global well-posedness of full compressible magnetohydrodynamic system in 3D bounded domains with large oscillations and vacuum. arXiv:2208.04480, Li et al. in Global existence of classical solutions to full compressible Navier–Stokes equations with large oscillations and vacuum in 3D bounded domains. arXiv:2207.00441), where the exponential growth of the gradient of density in L p -norm was explored. [ABSTRACT FROM AUTHOR]

We consider degenerate Kolmogorov-Fokker-Planck operators ℒu=∑i,j=1qaij(x,t)uxixj+∑k,j=1Nbjkxkuxj−ut,{\mathcal{ {\mathcal L} }}u=\mathop{\sum }\limits_{i,j=1}^{q}{a}_{ij}\left(x,t){u}_{{x}_{i}{x}_{j}}+\mathop{\sum }\limits_{k,j=1}^{N}{b}_{jk}{x}_{k}{u}_{{x}_{j}}-{u}_{t}, such that the corresponding model operator having constant aij{a}_{ij} is hypoelliptic, translation invariant w.r.t. a Lie group operation in RN+1{{\mathbb{R}}}^{N+1} and 2-homogeneous w.r.t. a family of nonisotropic dilations. We assume that the aij{a}_{ij}’s are globally bounded and Hölder continuous in xx (w.r.t. some intrinsic distance induced by ℒ{\mathcal{ {\mathcal L} }}); the matrix {aij}i,j=1q{\{{a}_{ij}\}}_{i,j=1}^{q} is symmetric and uniformly positive on Rq{{\mathbb{R}}}^{q}. We prove “partial Schauder a priori estimates” on a bounded open set Ω⊂RN+1\Omega \subset {{\mathbb{R}}}^{N+1}, of the kind ∑i,j=1q([uxixj]∗,α,2x+[uxixj]∗,2)≤c{[ℒu]∗,α,2x+[ℒu]∗,2+supΩ∣u∣}\mathop{\sum }\limits_{i,j=1}^{q}({\left[{u}_{{x}_{i}{x}_{j}}]}_{\ast ,\alpha ,2}^{x}+{\left[{u}_{{x}_{i}{x}_{j}}]}_{\ast ,2})\le c\left\{{\left[{\mathcal{ {\mathcal L} }}u]}_{\ast ,\alpha ,2}^{x}+{\left[{\mathcal{ {\mathcal L} }}u]}_{\ast ,2}+\mathop{\sup }\limits_{\Omega }| u| \right\} for suitable functions uu, where [u]∗,α,2x=sup(x,t)≠(y,t)∈Ωd(x,t),(y,t)2+α∣u(x,t)−u(y,t)∣‖x−y‖α[u]∗,2=supξ∈Ωdξ2∣u(ξ)∣.\begin{array}{rcl}{\left[u]}_{\ast ,\alpha ,2}^{x}& =& \mathop{\sup }\limits_{\left(x,t)\ne (y,t)\in \Omega }{d}_{\left(x,t),(y,t)}^{2+\alpha }\frac{| u\left(x,t)-u(y,t)| }{\Vert x-y{\Vert }^{\alpha }}\\ {{[}u]}_{\ast ,2}& =& \mathop{\sup }\limits_{\xi \in \Omega }{d}_{\xi }^{2}| u\left(\xi )| .\end{array} Here ‖⋅‖\Vert \cdot \Vert is a homogeneous norm in RN{{\mathbb{R}}}^{N}, while dξ=dist(ξ,∂Ω){d}_{\xi }={\rm{dist}}\left(\xi ,\partial \Omega ) and dξ,η=min{dξ,dη}{d}_{\xi ,\eta }=\min \left\{{d}_{\xi },{d}_{\eta }\right\}. We also prove that the derivatives uxixj{u}_{{x}_{i}{x}_{j}} are locally Hölder continuous in space and time.

Parametric finite element methods have achieved great success in approximating the evolution of surfaces under various different geometric flows, including mean curvature flow, Willmore flow, surface diffusion, and so on. However, the convergence of Dziuk's parametric finite element method, as well as many other widely used parametric finite element methods for these geometric flows, remains open. In this article, we introduce a new approach and a corresponding new framework for the analysis of parametric finite element approximations to surface evolution under geometric flows, by estimating the projected distance from the numerically computed surface to the exact surface, rather than estimating the distance between particle trajectories of the two surfaces as in the currently available numerical analyses. The new framework can recover some hidden geometric structures in geometric flows, such as the full H 1 parabolicity in mean curvature flow, which is used to prove the convergence of Dziuk's parametric finite element method with finite elements of degree k ≥ 3 for surfaces in the three-dimensional space. The new framework introduced in this article also provides a foundational mathematical tool for analyzing other geometric flows and other parametric finite element methods with artificial tangential motions to improve the mesh quality. [ABSTRACT FROM AUTHOR]

In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions à la Luckhaus–Sturzenhecker to such flows, the latter result holding in low dimension and conditionally to the convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows one to simplify many arguments. [ABSTRACT FROM AUTHOR]

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some contracting and convergence results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm–Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier–Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is the study of second-order optimality conditions for the bilinear control of a strongly degenerate parabolic equation. The equation is degenerate at the boundary of the spatial domain. The well-posedness of the state equation, as well as weak maximum principles are established. We prove some differentiability properties of the control-to-state operator and the existence of optimal solutions. Finally, we derive first- and second-order optimality conditions for the system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and regularity of a capacity solution for a coupled nonlinear anisotropic parabolic-elliptic system, where the elliptic component of the parabolic equation involves thermal conductivities $ a_{i}(u) $ a i (u) that satisfy $ \lim _{s\rightarrow +\infty }a_{i}(s)=0 $ lim s → + ∞ a i (s) = 0 for all $ i=1,\ldots,N $ i = 1 , ... , N. We work with anisotropic Sobolev spaces and use Schauder's fixed-point theorem to find weak solutions to approximate problems. In addition, we validate the convergence of a subsequence of approximate solutions to a capacity solution. [ABSTRACT FROM AUTHOR]

The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations. [ABSTRACT FROM AUTHOR]

In this paper, we show an abstract setting for analyzing the solvability of a class of nonlinear mixed degenerate parabolic equations. Specifically, we combine the well-known theory for stationary mixed problems and the theory for nonlinear degenerate parabolic equations, to obtain sufficient conditions inspired in real-world application problems to guarantee the existence and uniqueness of solution. Furthermore, we illustrate some applications of the abstract framework through particular problems arising from the field of physical electromagnetic sciences. Initially, we introduce two formulations based on a time primitive of the electric field, encompassing both local and nonlocal source conditions. Finally, we explore a nonlinear axisymmetric eddy current model employed in the context of induction furnaces. [ABSTRACT FROM AUTHOR]

Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation. [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]

In this paper we prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in L 1 (Ω) , there exists a solution that "instantly" becomes bounded. Moreover we study the behavior in time of these solutions showing that this class of problems admits global solutions which all have the same behavior in time independently of the size of the initial data. [ABSTRACT FROM AUTHOR]

We are concerned with a class of degenerate diffusion equations with time delay describing population dynamics with age structure. In our recent study [Nonlinearity, 33 (2020), 4013–4029], we established the existence and uniqueness of critical traveling wave for the time-delayed degenerate diffusion equations, and obtained the reducing mechanism of time delay on critical wave speed. In this paper, we now are able to show the asymptotic spreading speed and its coincidence with the critical wave speed c ∗ (m , r) of sharp wave, and prove that the initial perturbation or the boundary of the compact support of the solution propagates at the critical wave speed c ∗ (m , r) for the time-delayed degenerate diffusion equations. Remarkably, different from the existing studies related to spreading speeds, the time delay and the degenerate diffusion lead to some essential difficulties in the analysis of the spreading speed, because the time delay makes the critical speed of traveling waves slow down in a more complicated fashion such that the critical speed cannot be determined by the characteristic equation, and the degenerate diffusion causes the loss of regularity for the solutions. By a phase transform technique we construct upper and lower solutions with semi-compact supports and then we determine the asymptotic spreading speed. Furthermore, we propose a brand-new sharp-profile-based difference scheme to handle large variation of degenerate diffusion (u m) xx near the sharp edge and carry out some numerical simulations which perfectly confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In a ball Ω = B R (0) ⊂ R n , n ≥ 2 , the chemotaxis system u t = ∇ · (D (u) ∇ u) - ∇ · (u S (u) ∇ v) , 0 = Δ v - μ + u , μ = 1 | Ω | ∫ Ω u , (⋆) is considered under no-flux boundary conditions, with a focus on nonlinearities S ∈ C 2 ([ 0 , ∞)) which exhibit super-algebraically fast decay in the sense that with some K S > 0 , β ∈ [ 0 , 1) and ξ 0 > 0 , S (ξ) > 0 and S ′ (ξ) ≤ - K S ξ - β S (ξ) for all ξ ≥ ξ 0. It is, inter alia, shown that if furthermore D ∈ C 2 ((0 , ∞)) is positive and suitably small in relation to S by satisfying ξ S (ξ) D (ξ) ≥ K SD ξ λ for all ξ ≥ ξ 0 with some K SD > 0 and λ > 2 n , then throughout a considerably large set of initial data, (⋆ ) admits global classical solutions (u, v) fulfilling z (t) C ≤ ‖ u (· , t) ‖ L ∞ (Ω) ≤ C z (t) for all t > 0 , with some C = C (u , v) ≥ 1 , where z denotes the solution of z ′ (t) = z 2 (t) · S (z (t)) , t > 0 , z (0) = ξ 0 , which is seen to exist globally, and to satisfy z (t) → + ∞ as t → ∞ . As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in (⋆ ) at logarithmic and doubly logarithmic rates, respectively. [ABSTRACT FROM AUTHOR]

We study local boundedness and Hölder continuity of a parabolic equation involving the fractional p-Laplacian of order s, with 0 < s < 1 , 2 ≤ p < ∞ , with a general right-hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right-hand side. [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]

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]

In this paper, we prove a new blowup criterion for the strong solution to the Cauchy problem of three-dimensional micropolar fluid equation with vacuum. Specifically, we establish a blowup criterion in terms of L t ∞ L x q of the density, where 1 < q < ∞ , and it is independent on the velocity of rotation of the microscopic particles. [ABSTRACT FROM AUTHOR]

We obtain weighted Poincaré inequalities in bounded domains, where the weight is given by a symmetric nonnegative definite matrix, which can degenerate on submanifolds. Furthermore, we investigate uniqueness and nonuniqueness of solutions to degenerate elliptic and parabolic problems, where the diffusion matrix can degenerate on subsets of the boundary of the domain. Both the results are obtained by means of the distance function from the degeneracy set, which is used to construct suitable local sub– and supersolutions. [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 article, we obtain Hessian estimates for Kolmogorov–Fokker–Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in harmonic analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative Hessian estimates for the Kolmogorov–Fokker–Planck operators. [ABSTRACT FROM AUTHOR]

In this paper, we derive a comparison principle for non-negative weak sub- and super-solutions to doubly nonlinear parabolic partial differential equations whose prototype is ∂ t u q - div (| ∇ u | p - 2 ∇ u) = 0 in Ω T , with q > 0 and p > 1 and Ω T : = Ω × (0 , T) ⊂ R n + 1 . Instead of requiring a lower bound for the sub- or super-solutions in the whole domain Ω T , we only assume the lateral boundary data to be strictly positive. The main results yield some applications. Firstly, we obtain uniqueness of non-negative weak solutions to the associated Cauchy–Dirichlet problem. Secondly, we prove that any weak solution is also a viscosity solution. [ABSTRACT FROM AUTHOR]

In this paper we conceive and analyze a discrete duality finite volume (DDFV) scheme for the unsteady generalized thermistor problem, including a p-Laplacian for the diffusion and a Joule heating source. As in the continuous setting, the main difficulty in the design of the discrete model comes from the Joule heating term. To cope with this issue, the Joule heating term is replaced with an equivalent key formulation on which a fully implicit scheme is constructed. Introducing a tricky cut-off function in the proposed discretization, we are able to recover the energy estimates on the discrete temperature. Another feature of this approach is that we dispense with the discrete maximum principle on the approximate electric potential, which in essence poses restrictive constraints on the mesh shape. Then, the existence of discrete solution to the coupled scheme is established. Compactness estimates are also shown. Under general assumptions on the data and meshes, the convergence of the numerical scheme is addressed. Numerical results are finally presented to show the efficiency and accuracy of the proposed methodology as well as the behavior of the implemented nonlinear solver. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following quasilinear chemotaxis system involving nonlocal effect u t = ∇ · (φ (u) ∇ u) - ∇ · (u ∇ v) + μ u 1 - ∫ Ω u α d x , x ∈ Ω , t > 0 , [ 2.5 m m ] 0 = Δ v - m (t) + u , m (t) = 1 | Ω | ∫ Ω u (x , t) d x , x ∈ Ω , t > 0 , [ 2.5 m m ] u (x , 0) = u 0 (x) , x ∈ Ω , where Ω = B R (0) ⊂ R n (n ≥ 3) with R > 0 , the parameters μ , α are positive constants and diffusion function φ (u) ≤ C 0 (1 + u) - m for all u ≥ 0 with C 0 > 0 and m > - 1. It has been shown that if 0 < α < min 2 , n 2 , n (m + 1) 2 , then there exist suitable initial data u 0 such that the corresponding radially symmetric solution blows up in finite time. In this work, we extend the blow-up result established by previous researchers. [ABSTRACT FROM AUTHOR]

We study the no-flux initial-boundary value problem of a chemotaxis system with singular sensitivity of the following form ⋆ u t = Δ u - χ ∇ · u ∇ ln v - u + g 1 , v t = Δ v - v + u v (1 - v) + g 2 , over a bounded domain Ω ⊂ R n , with chemotaxis coefficient χ > 0 and nonnegative source functions g 1 and g 2 , which was proposed by Pitcher to describe the dynamics of burglaries. From the recent results it is known that, if n = 2 , then such problem admits a global generalized solution for any initial data and any χ > 0 , and possesses a global classical solution for small initial data and small χ . This paper presents that for all reasonably regular initial data and any χ > 0 the corresponding homogeneous Neumann initial-boundary value problem possesses a generalized solution in higher-dimensional settings (i.e., n ≥ 3 ). The asymptotic behavior of generalized solutions is explored as well under some additional assumptions on the source functions. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem for degenerate parabolic and pseudo-parabolic equations associated with Hörmander-type operator. Under the subcritical growth restrictions on the nonlinearity f(u), which are determined by the generalized Métivier index, we establish the global existence of solutions and the corresponding attractors. Finally, we show the upper semicontinuity of the attractors in the topology of H X , 0 1 (Ω) . [ABSTRACT FROM AUTHOR]