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We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case where the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem understood in the sense of Chen–Perthame is both an entropy sub- and supersolution. It is established that the maximum of entropy subsolutions of the Cauchy problem is also an entropy subsolution of this problem. This result is used to prove the existence of the largest entropy subsolution (and the smallest entropy supersolution). It is also shown that the largest entropy subsolution and the smallest entropy supersolution are also entropy solutions. [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]

Solutions of the nonlinear degenerate parabolic equations may be discontinuous. The classical numerical schemes cannot well handle such discontinuities. In this paper, a new sixth-order weighted compact nonlinear scheme for this type of equations is presented. This presented scheme consists of a sixth-order spatially implicit (or compact) or explicit central differencing for the second spatial derivatives at grid nodes and a weighted nonlinear interpolation technique for the first spatial derivatives at cell centers to reduce spurious oscillations near shock waves and discontinuities. Stability analysis of the sixth-order explicit weighted compact nonlinear scheme along with a total variation diminishing Runge–Kutta method for time discretization is given. Numerical results show that the presented scheme can attain the sixth-order accuracy in smooth regions and the non-oscillatory property in capturing discontinuities. [ABSTRACT FROM AUTHOR]

This paper presents a full classification of the short-time behavior of the solution and the interfaces in the Cauchy problem for the nonlinear second order singular parabolic PDE u t - Δ u m + b u β = 0 , x ∈ R N , 0 < t < T with nonnegative initial function u 0 such that s u p p u 0 = { | x | < R } , u 0 ∼ C (R - | x |) α , as | x | → R - 0 , where 0 < m < 1 , b , β , C , α > 0 . Depending on the relative strength of the fast diffusion and absorption terms the problem may have infinite ( β ≥ m ) or finite ( β < m ) speed of propagation. In the latter case, the interface surface t = η (x) may shrink, expand or remain stationary depending on the relative strength of the fast diffusion and strong absorption terms near the boundary of support, expressed in terms of the parameters m , β , α , and C. In all cases we prove the existence or non-existence of the interfaces, explicit formula for the interface asymptotics, and local solution near the interface or at infinity. [ABSTRACT FROM AUTHOR]

In this work, we prove the existence of a weak solutions for the initial boundary value problem associated with the nonlinear degenerate parabolic equations ∂u... div b(x, t, u, Vu) = φ(x, t) + div a(x, t, Vu). ∂t We will use the Topological degree theory for operators of the type T + S, where S is a bounded demicontinuous map of type (S+) and T is a linear densely defined maximal monotone map with respect to a domain of T, and we study this problem in the space Lp(0, T; Wj,p(Ω, ω)), where Ω is a bounded domain in RN (N ≥ 2), p≥2and ω isavectorofweightfunctions. [ABSTRACT FROM AUTHOR]

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency. [ABSTRACT FROM AUTHOR]

Two high order multi-augmented and improved augmented finite volume methods are proposed for solving nonlinear degenerate parabolic problems. The solution is represented as Puiseux series expansion in a subdomain with singularity, but contains undetermined parameters called augmented variables. The equation in regular subdomain is treated with high accuracy numerical methods and the unknown parameters can be solved simultaneously from the corresponding nonlinear system. The outstanding advantages of the proposed methods are that the degeneracy can be depicted by the semi-analytic solution, and we can get high order results globally. Specially, the convergence order for nonlinear degenerate parabolic problems is determined by the numerical schemes on regular subdomain, and the augmented methods have good robustness for solving degenerate or singular problems. Numerical examples for some degenerate parabolic equations confirm the efficiency of the new methods including the second and fourth order schemes. In particular, a two-dimensional singular elliptic equation with corner degeneracy is presented in the numerical experiments to demonstrate that the proposed methods can be extended to higher dimensional degenerate problems. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time. [ABSTRACT FROM AUTHOR]

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian-type reaction-diffusion equation of non-Newtonian elastic filtration $$u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1 \lt p \lt 2, \beta \gt 0.$$ If the interface is finite, it may expand, shrink or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In some range of parameters, strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are based on nonlinear scaling laws and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. A full description of small-time behaviour of the interfaces and local solutions near the interfaces for slow diffusion case when p>2 is presented in a recent paper by Abdulla and Jeli [(2017) Europ. J. Appl. Math.28(5), 827–853]. [ABSTRACT FROM AUTHOR]

This paper presents a full classification of the short-time behavior of the interfaces in the Cauchy problem for the nonlinear second order degenerate parabolic PDE u t - Δ u m + b u β = 0 , x ∈ R N , 0 < t < T with nonnegative initial function u 0 such that s u p p u 0 = { | x | < R } , u 0 ∼ C (R - | x |) α , as | x | → R - 0 , where m > 1 , C , α , β > 0 , b ∈ R . Interface surface t = η (x) may shrink, expand or remain stationary depending on the relative strength of the diffusion and reaction terms near the boundary of support, expressed in terms of the parameters m , β , α , s i g n b and C. In all cases we prove explicit formula for the interface asymptotics, and local solution near the interface. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear parabolic problems including the p-Laplacian equation. The initial datum and the forcing term are allowed to be summable functions or Radon measures. We prove that these equations have surprising regularizing properties. Moreover, we study the behavior in time of these solutions proving that decay estimates hold true also for nonzero reaction terms. Finally, we study the autonomous case. [ABSTRACT FROM AUTHOR]

Numerical discretizations of the Generalized Porous Medium Equation (GPME) with discontinuous coefficients are analyzed with respect to the formation of numerical artifacts. In addition to the degeneracy and self-sharpening of the GPME with continuous coefficients, detailed in [1] , increased numerical challenges occur in the discontinuous coefficients case. These numerical challenges manifest themselves in spurious temporal oscillations in second order finite volume discretizations with both arithmetic and harmonic averaging. The integral average, developed in [2] , leads to improved solutions with monotone and reduced amplitude temporal oscillations. In this paper, we propose a new method called the Shock-Based Averaging Method (SAM) that incorporates the shock position into the numerical scheme. The shock position is numerically calculated by discretizing the theoretical speed of the front from the GPME theory. The speed satisfies the jump condition for integral conservation laws. SAM results in a non-oscillatory temporal profile, producing physically valid numerical results. We use SAM to demonstrate that the choice of averaging alone is not the cause of the oscillations, and that the shock position must be a part of the numerical scheme to avoid the artifacts. [ABSTRACT FROM AUTHOR]

In this article, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long-time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all time is established in a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality-type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates. [ABSTRACT FROM AUTHOR]

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in [ABSTRACT FROM AUTHOR]

We study the expanded mixed finite element method applied to degenerate parabolic equations with the Dirichlet boundary condition. The equation is considered a prototype of the nonlinear Forchheimer equation for slightly compressible fluid flow in porous media. The bounds for the solutions are established. In both continuous and discrete time procedures, utilizing the monotonicity properties of Forchheimer equation and the boundedness of solutions, we establish the error estimates in L 2 -norm for solutions, divergence of the vector variable in several Lebesgue norms. A numerical example using the lowest order Raviart–Thomas ( R T 0 ) mixed element confirms our theoretical results regarding convergence rates. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonexistence and longtime behavior of weak solution for the degenerate parabolic equation ∂ u = u div(|∇ u |∇ u ) + γ|∇ u | + β u with zero boundary condition. Blow-up time is derived when the blow-up does occur. [ABSTRACT FROM AUTHOR]

We prove existence and uniqueness of entropy solutions for the Cauchy problem of weakly coupled systems of nonlinear degenerate parabolic equations. We prove existence of an entropy solution by demonstrating that the Engquist-Osher finite difference scheme is convergent and that any limit function satisfies the entropy condition. The convergence proof is based on deriving a series of a priori estimates and using a general $L^p$ compactness criterion. The uniqueness proof is an adaption of Kruzkov's ``doubling of variables'' proof. We also present a numerical example motivated by biodegradation in porous media.

The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids, and then study the expanded mixed finite element method applied to the initial boundary value problem for the resulting degenerate parabolic equation for pressure. The bounds for the solutions, time derivative, and gradient of solutions are established. Utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions, a priori error estimates for solution are obtained in L²-norm, L∞-norm as well as for its gradient in L2-a-norm for all a (0, 1). Optimal L²-error estimates are shown for solutions under some additional regularity assumptions. Numerical results using the lowest order Raviart-Thomas mixed element confirm the theoretical analysis regarding convergence rates. [ABSTRACT FROM AUTHOR]

We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some “asymmetric systems” where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations using a generalization of the strong maximum principle for systems. [ABSTRACT FROM AUTHOR]

We prove the existence of a Radon measure-valued solution for a class of nonlinear degenerate parabolic equations with a 'logarithmic diffusion' when the initial datum u is a bounded Radon measure, and we study the regularity of these solutions. In particular, we prove that a regularizing effect appears if the initial datum is diffused with respect to the ' C-capacity' since in this case the solution becomes a summable function. Finally, we study the uniqueness of these measure-valued solutions. [ABSTRACT FROM AUTHOR]

In this paper, we develop implicit finite difference schemes with a semi-implicit weighted essentially non-oscillatory (WENO) reconstruction for nonlinear degenerate parabolic equations. Such degenerate equations would exist finite speed front propagation which is similar to discontinuous solutions for hyperbolic equations. We propose to use the finite difference WENO scheme with an implicit time discretization, to avoid the parabolic time step condition from an explicit time discretization, namely Δ t = O (Δ x 2) where Δ x is the mesh size. We further simplify the resulting nonlinear system by employing a semi-implicit WENO reconstruction, that is, nonlinear weights are treated explicitly while keeping linear reconstructions of derivatives implicitly. The novelty is that the semi-implicit WENO reconstruction maintains the original high order accuracy both in space and in time, as well as unconditional stability as a fully implicit scheme, but the scheme is much simpler in complexity than the fully implicit one. Numerical experiments are performed to show the high order accuracy, large time step conditions with better efficiency as compared to an explicit scheme, and good performances for capturing front propagation. • A semi-implicit WENO reconstruction for nonlinear parabolic equations is proposed. • Two different formulae of FD WENO schemes for parabolic equations are discussed. • The semi-implicit WENO approach keeps the original high order temporal accuracy. • The new approach maintains an unconditional stability as a fully implicit scheme. • Numerical experiments demonstrate the large time step condition and high efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials: in a cylinder Ω × (0, T) with initial condition u ( z, 0) = u ( z) ≥ 0 and vanishing on the boundary ∂Ω × (0, T), where Ω is a Carnot-Carathéodory metric ball in ℝ and the time-dependent singular potential function is V ( z, t) ∈ L (Ω × (0, T)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles-Souganidis (2000) for first-order Hamilton-Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations. [ABSTRACT FROM AUTHOR]

We investigate the behavior for large times of nonnegative solutions to the Dirichlet problem in cone-like domains for the porous media equation. We obtain optimal estimates for the sup norm of the solution and for the size of its support. We also consider the case where a damping term depending on the space gradient of the solution appears. In this case we also identify the critical behavior of the damping term discriminating between decay to zero of a suitable moment of the solution as t → + ∞, and stabilization of the moment to a positive constant. [ABSTRACT FROM AUTHOR]

We prove that weak limits of approximate entropy solutions to a one-dimensional degenerate parabolic equation are entropy solutions as well. [ABSTRACT FROM AUTHOR]

In this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computational cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its convergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution. [ABSTRACT FROM AUTHOR]

This paper concerns a nonlinear doubly degenerate reaction–diffusion equation which appears in a bacterial growth model and is also of considerable mathematical interest. A travelling wave analysis for the equation is carried out. In particular, the qualitative behaviour of both sharp and smooth travelling wave solutions is analysed. This travelling wave behaviour is also verified by some numerical computations for a special case. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we study the existence of solution to nonlinear degenerate parabolic equations with measure data. [Copyright &y& Elsevier]

We prove existence and uniqueness of entropy solutions for the Cauchy problem of weakly coupled systems of nonlinear degenerate parabolic equations. We prove existence of an entropy solution by demonstrating that the Engquist-Osher finite difference scheme is convergent and that any limit function satisfies the entropy condition. The convergence proof is based on deriving a series of a priori estimates and using a general Lp compactness criterion. The uniqueness proof is an adaption of Kruzkov's "doubling of variables" proof. We also present a numerical example motivated by biodegradation in porous media. [ABSTRACT FROM AUTHOR]

The evolution of interfaces and the local behavior of solutions near the interface in problems for one-dimensional reaction-diffusion equations are studied. In all cases explicit formulae for the interface, with accuracy up to constant coefficients, are found together with the local solution. The methods used are matched asymptotic expansions for preliminary formal results, and rescaling and a barrier technique for rigorous proof, using special comparison theorems in irregular domains. [ABSTRACT FROM AUTHOR]