Nonlinear Degenerate Equations Research Articles

Fully nonlinear degenerate equations with applications to Grad equations

Fully nonlinear degenerate equations with applications to Grad equations

High order hybrid asymptotic augmented finite volume methods for nonlinear degenerate wave equations

High order hybrid asymptotic augmented finite volume methods for nonlinear degenerate wave equations

Mixed boundary value problems for fully nonlinear degenerate or singular equations

We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Hölder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Hölder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems.

Solvability of an initial-boundary value problem for a nonlinear pseudoparabolic equation with degeneration

This article is devoted to the solvability of degenerate nonlinear equations of pseudoparabolic type. Such problems appear naturally in physical and biological models. The article aims to study the solvability in the classes of regular solutions of (all derivatives generalized in the sense of S.L. Sobolev included in the equation) initial-boundary value problems for differential equations. For the problems under consideration, we have found conditions on parameters ensuring the existence of solutions and we have proved existence and uniqueness theorems. The main method for proving the solvability of boundary value problems is the regularization method.

Mixed control for degenerate nonlinear equations with fractional derivatives

Mixed control for degenerate nonlinear equations with fractional derivatives

Fully nonlinear degenerate equations with sublinear gradient term

We establish existence and uniqueness of positive viscosity solutions of Pk±(D2u)+|Du|qup=0inΩ,u=0on∂Ω,where k<N,Ω is a bounded domain in RN,N≥2,0<p<1,0≤q<1 and Pk± are degenerate elliptic operators. First, we use of a change of dependent variable originating in Brezis and Kamin (1992) in order to convert the equation into one with the right monotonicity in the u-variable. Thereafter by applying Perron’s method, we prove the existence and uniqueness of the solutions. Using an a-priori estimate, we show the nonexistence of subsolutions. We also find the ranges of p and q for the existence and nonexistence results.

Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science

We consider a nonlinear degenerate reaction-diffusion equation. First we prove that if the initial state is nonnegative, then the solution remains nonnegative for all time. Then we prove the approximate controllability between nonnegative states via multiplicative controls, this is done using the reaction coefficient as control.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/59/abstr.html

General decay and blowup of solutions for a degenerate viscoelastic equation of Kirchhoff type with source term

In this work, we consider a nonlinear degenerate viscoelastic equation of Kirchhoff type with source term in a bounded domain. Under suitable assumptions on the relaxation functions g and M, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow-up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.

УЗАГАЛЬНЕНИЙ РОЗВ'ЯЗОК ЗАДАЧІ ДІРІХЛЕ ДЛЯ МОДЕЛЬНОГО АНІЗОТРОПНОГО ВАГОВОГО РІВНЯННЯ

The paper deals with the Dirichlet problem for a model nonlinear degenerate anisotropic elliptic second-order equation. Anisotropy and degeneration (with respect to the independent variables) is characterized by the presence of different exponents q1 , q2 and weighted functions |x|^q1 та |x|^q2 in the left side of the equation. The main result of the paper is theorem on the existence of the generalized solution of the Dirichlet problem under consideration.

Escobar type theorems for elliptic fully nonlinear degenerate equations

In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also consider non-degenerate equations. Such subdomains are geodesic balls in Sm, punctured balls and annular domains. Our results extend those of Escobar when m≥ 3, and Hang-Wang and Jimenez when m= 2.

Travelling breathers and solitary waves in strongly nonlinear lattices.

We study the existence of travelling breathers and solitary waves in the discrete p-Schrödinger (DpS) equation. This model consists of a one-dimensional discrete nonlinear Schrödinger (NLS) equation with strongly nonlinear inter-site coupling (a discrete p-Laplacian). The DpS equation describes the slow modulation in time of small amplitude oscillations in different types of nonlinear lattices, where linear oscillators are coupled to nearest-neighbours by strong nonlinearities. Such systems include granular chains made of discrete elements interacting through a Hertzian potential (p = 5/2 for contacting spheres), with additional local potentials or resonators inducing local oscillations. We formally derive three amplitude PDEs from the DpS equation when the exponent of nonlinearity is close to (and above) unity, i.e. for p lying slightly above 2. Each model admits localized solutions approximating travelling breather solutions of the DpS equation. One model is the logarithmic NLS equation which admits Gaussian solutions, and the other is fully nonlinear degenerate NLS equations with compacton solutions. We compare these analytical approximations to travelling breather solutions computed numerically by an iterative method, and check the convergence of the approximations when [Formula: see text] An extensive numerical exploration of travelling breather profiles for p = 5/2 suggests that these solutions are generally superposed on small amplitude non-vanishing oscillatory tails, except for particular parameter values where they become close to strictly localized solitary waves. In a vibro-impact limit where the parameter p becomes large, we compute an analytical approximation of solitary wave solutions of the DpS equation.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

The Cauchy problem for a class of nonlinear degenerate parabolic-hyperbolic equations

In this article, we prove a general existence theorem for a class of nonlinear degenerate parabolic-hyperbolic equations. Since the regions of parabolicity and hyperbolicity are coupled in a way that depends on the solution itself, there is almost no hope of decoupling the regions and then taking into account the parabolic and the hyperbolic features separately. The existence of solutions can be obtained by finding the limit of solutions for the regularized equation of strictly parabolic type. We use the energy methods and vanishing viscosity methods to prove the local existence and uniqueness of solution.

Regularity Properties of Degenerate Diffusion Equations with Drifts

This paper considers a class of nonlinear degenerate drift-diffusion equations. We study well-posedness and regularity properties of the solutions with the goal of achieving uniform Holder regulari...

Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations.

This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h′(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1) h′(u) is constant k, (2) h′(u) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.

On a Nonlinear Degenerate Evolution Equation with Nonlinear Boundary Damping

This paper deals essentially with a nonlinear degenerate evolution equation of the formKu″-Δu+∑j=1nbj∂u′/∂xj+uσu=0supplemented with nonlinear boundary conditions of Neumann type given by∂u/∂ν+h·, u′=0. Under suitable conditions the existence and uniqueness of solutions are shown and that the boundary damping produces a uniform global stability of the corresponding solutions.

Entropy estimates of radial symmetric solutions to a fourth order nonlinear degenerate equation in higher dimensions

Entropy estimates of radial symmetric solutions to a fourth order nonlinear degenerate equation in higher dimensions

Nonlinear degenerate parabolic equations with time-dependent singular potentials for Baouendi-Grushin vector fields

In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials: $$\begin{array}{*{20}c} {\frac{{\partial u^q }} {{\partial t}} = \nabla _\alpha \cdot \left( {\left\| z \right\|^{ - p\gamma } \left| {\nabla _\alpha u} \right|^{p - 2} \nabla _\alpha u} \right) + V(z,t)u^{p - 1} ,} \\ {\frac{{\partial u^q }} {{\partial t}} = \nabla _\alpha \cdot \left( {\left\| z \right\|^{ - 2\gamma } \nabla _\alpha u^m } \right) + V(z,t)u^m ,} \\ {\frac{{\partial u^q }} {{\partial t}} = u^\mu \nabla _\alpha \cdot \left( {u^\tau \left| {\nabla _\alpha u} \right|^{p - 2} \nabla _\alpha u} \right) + V(z,t)u^{p - 1 + \mu + \tau } } \\ \end{array}$$ in a cylinder Ω × (0, T) with initial condition u (z, 0) = u0 (z) ≥ 0 and vanishing on the boundary ∂Ω × (0, T), where Ω is a Carnot-Caratheodory metric ball in ℝd+k and the time-dependent singular potential function is V (z, t) ∈ Lloc1 (Ω × (0, T)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.

A Fitted Finite Volume Method for Unit-linked Policy with Surrender Option

In this paper, fitted finite volume method is developed to solve a nonlinear degenerate Black-Scholes equation applied in the valuation of unit-linked policy with surrender option, based on the fitting idea in S. Wang [IMA J. Numer. Anal., 24 (2004), 699--720]. Unlike the conventional pricing method mentioned in [1] which is using the free boundary method to calibrate the valuation PDE, here we develop a power penalty method to solve numerically the linear complimentary problem in the variational inequality arising from the valuation of unit-linked policy with surrender option. With the degenerate boundary and non-smooth final condition, we will show that it is essential to refine the mesh to remain the convergence and super-convergence order.

ENTROPY ESTIMATES AND LARGE-TIME BEHAVIOR OF SOLUTIONS TO A FOURTH-ORDER NONLINEAR DEGENERATE EQUATION

The main aim of this paper is to give some theoretical analysis for a nonlinear fourth-order degenerate equation related to image processing, with homogeneous Neumann and no-flux boundary conditions. The asymptotic behavior of solutions to the equation is discussed in the paper using the entropy dissipation method. We firstly derive some entropy estimates via the algebraic approach, and present a polynomial decay of an entropy. As a result, we prove solutions of the equation eventually converge to equilibrium, which can help us understand the model in greater depth. To the best of our knowledge, this is the first result on the long-time behavior of the fourth-order model in image processing.

Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge–Ampère type

Abstract. We study fully nonlinear partial differential equations of Monge–Ampère type involving the derivatives with respect to a family of vector fields. The main result is a comparison principle among viscosity subsolutions, convex with respect to , and viscosity supersolutions (in a weaker sense than usual), which implies the uniqueness of solution to the Dirichlet problem. Its assumptions include the equation of prescribed horizontal Gauss curvature in Carnot groups. By the Perron method we also prove the existence of a solution either under a growth condition of the nonlinearity with respect to the gradient of the solution, or assuming the existence of a subsolution attaining continuously the boundary data, therefore generalizing some classical result for Euclidean Monge–Ampère equations.