Quasilinear Parabolic Differential Equations Research Articles

Quasi linear parabolic pde posed on a network with non linear Neumann boundary condition at vertices

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.

Implicit difference schemes for quasilinear parabolic functional equations

AbstractWe present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.

Stability of difference problems generated by infinite systems of quasilinear parabolic functional differential equations

The paper deals with infinite weakly coupled systems of quasilinear parabolic differential functional equations. Initial boundary conditions of the Robin type are considered. We construct an explicit Euler type approximation method based on an infinite system of difference functional equations. Next we apply the truncation method to obtain a finite difference scheme corresponding to the original differential problem. We present a complete convergence analysis for the methods. The results are based on a comparison technique with nonlinear estimates of the Perron type for given functions.

Difference methods for infinite systems of quasilinear parabolic functional differential equations

AbstractClassical solutions of initial boundary value problems for infinite systems of quasilinear parabolic differential functional equations are considered. Two type of difference schemes are constructed. We prove that solutions of infinite difference schemes approximate solutions of our differential functional problem. In the second part of the paper we show that solutions of infinite differential functional systems can be approximated by solutions of difference systems with initial boundary conditions and the systems are finite.A complete convergence analysis for the methods is presented. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions.

Stability of spheres under volume-preserving mean curvature flow

We give a new, elementary proof of the theorem, due to J. Escher and G. Simonett, that for the initial conditions close to Eucleadian spheres the solutions of the volume-preserving mean curvature flow converge to Eucleadian spheres (which, in general, differ from the initial spheres). Our result is in the metric given by Sobolev norms. While the proof by J. Escher and G. Simonett uses extensively rather involved results from the infinite-dimensional invariant manifold theory and quasilinear parabolic differential equations, our main point is to use an orthogonal decomposition of the solutions near the manifold of Euclidean spheres and differential inequalities for the Lyapunov functionals. Apart from local well-posedness, which is proven along standard lines, our proof is completely self-contained.

DIFFERENCE METHODS FOR INFINITE SYSTEMS OF QUASILINEAR PARABOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS

Classical solutions of initial boundary value problems for infinite systems of quasilinear parabolic differential functional equations are considered. Two type of difference schemes are constructed. We prove that solutions of infinite difference schemes approximate solutions of our differential functional problem. In the second part of the paper we show that solutions of infinite differential functional systems can be approximated by solutions of difference systems with initial boundary conditions and the systems are finite. A complete convergence analysis for the methods is presented. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions. Mathematics Subjects Classification: 65M10, 35K50, 35R10.

On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics

Quasilinear parabolic functional differential equations containing multiple transformations of spatial variables are considered with the Neumann boundary-value conditions. Sufficient conditions of the Andronov-Hopf bifurcation of periodic solutions are obtained along with expansions of the solutions in powers of a small parameter. Spectral properties of the linearized elliptic operator of this problem are investigated. Necessary and sufficient conditions of normality are obtained for such operators. Examples illustrating their properties are given.

Harnack estimates for quasi-linear degenerate parabolic differential equations

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Holder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.

Laguerre spectral approximation of Stokes’ first problem for third-grade fluid

A Laguerre–Galerkin method is proposed and analyzed for Quasilinear parabolic differential equation which arises from Stokes’ first problem for a third-grade fluid on a semi-infinite interval. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations to the transformed equations is developed and implemented. Effects of non-Newtonian parameters on the flow phenomena are analyzed and documented.

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i ( u i ) may have the property D i ( 0 ) = 0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.

On a system of quasilinear parabolic functional differential equations

We consider initial boundary value problems for a system of second order quasilinear parabolic equations where also the main part contains functional dependence on the unknown function. This system is of type, considered in [6], [7] by U. Hornung, W. Jager and A. Mikelic.

On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs

We study the Cauchy–Dirichlet problem for a second-order quasilinear parabolic stochastic differential equation (SPDE) in a domain with a zero order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Holder classes with weights.

Implicit difference methods for parabolic functional differential equations

AbstractWe present a new class of numerical methods for the solution of quasilinear parabolic functional differential equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that the new methods are considerable better that the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators. Results obtained in the paper can be applied to differential integral problems and equations with retarded variables.

Strongly nonlinear impulsive evolution equations and optimal control

Strongly nonlinear impulsive evolution equations are investigated. Existence of solutions of strongly nonlinear impulsive equations is proved and some properties of the solutions are discussed. These results are applied to Lagrange problems of optimal control and we proved existence results. For illustration, an example of a quasi-linear impulsive parabolic differential equation and the corresponding optimal control is also presented.

Numerical approximations of parabolic differential functional equations with the initial boundary conditions of the Neumann type

The aim of this paper is to present a numerical approximation for quasilinear parabolic differential functional equations with initial boundary conditions of the Neumann type. The convergence result is proved for a difference scheme with the property that

A comparison analysis of Sivashinsky's type evolution equations describing flame propagation in channels

We establish a comparison between Rakib–Sivashinsky and Michelson–Sivashinsky quasilinear parabolic differential equations governing the weak thermal limit of flame front propagating in channels. For the former equation, we give a complete description of all steady solutions and present their local and global stability analysis. For the latter, bi-coalescent and interpolating unstable steady solutions are introduced and shown to be more numerous than the previous known coalescent solutions. These facts are argued to be responsible for the disagreement between the observed dynamics in numerical experiments and the exact (linear) stability analysis and give ingredients to construct quasi-stable solutions describing parabolic steadily propagating flame with centered tip.

On periodic solutions of nonlinear evolution equations in Banach spaces

We prove an existence result for T-periodic solutions to nonlinear evolution equations of the form x ̇ (t)+A t,x(t) =f t,x(t) , 0<t<T. Here V↪H↪V ∗ is an evolution triple, A :I×V→V ∗ is a uniformly monotone operator, and f :I×H→V ∗ is a Caratheodory mapping which is Hölder continuous with respect to x in H and exponent 0< α⩽1. For illustration, an example of a quasi-linear parabolic differential equation is worked out in detail.

On the Cauchy Problem for Stochastic Stokes Equations

We extend Krylov's Lp-solvability theory of the second order quasi-linear parabolic stochastic differential equations to the stochastic Stokes equation. Some additional integrability and regularity properties are also presented.

Time-Periodic Solutions of Quasilinear Parabolic Differential Equations: I. Dirichlet Boundary Conditions

We study boundary value problems for quasilinear parabolic equations when the initial condition is replaced by periodicity in the time variable. Our approach is to relate the theory of such problems to the classical theory for initial-boundary value problems. In the process, we generalize many previously known results.

Positive and Probabilistic Solutions of One-Dimensional Quasilinear Parabolic Differential Equations

Positive and Probabilistic Solutions of One-Dimensional Quasilinear Parabolic Differential Equations