Quasilinear Parabolic Problems Research Articles

A quasilinear parabolic model for population evolution

A quasilinear parabolic problem is investigated. It models the evolution of a single population species with a nonlinear diffusion and a logistic reaction function. We present a new treatment combining standard theory of monotone operators in L 2 (Ω) with some order- preserving properties of the evolutionary equation. The advantage of our approach is that we are able to obtain the existence and long-time asymptotic behavior of a weak solution almost simultaneously. We do not employ any uniqueness results; we rely on the uniqueness of the minimal and maximal solutions instead. At last, we answer the question of (long-time) survival of the population in terms of a critical value of a spectral parameter.

On some anisotropic, nonlocal, parabolic singular perturbations problems

This article is devoted to the study of the anisotropic singular perturbations theory for some quasilinear parabolic problems. Describing the asymptotic behaviour of the solutions of nonlocal problems yields to show the existence of solution to some integro-differential problems. The closeness between the solutions of these two kinds of problems is established.

Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω e that is e-periodically perforated by small holes of order . The holes are divided into three e-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure.

Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources

Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources

Symmetry of Positive Solutions of Asymptotically Symmetric Parabolic Problems on $${{\mathbb R}^N}$$

In this paper we investigate symmetry properties of positive solution of quasilinear parabolic problems in the whole space. As the main result, we prove that if the problem converges exponentially to a symmetric one, then the solution converges to the space of symmetric functions. We also show, that this result does not hold true, if the convergence is not exponential.

Asymptotic Symmetry for a Class of Quasi-Linear Parabolic Problems

Abstract We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the ω limit set are nonnegative radially symmetric solutions of the stationary problem.

Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition

This note deals with a class of heat emission processes in a medium with a non-negative source, a nonlinear decreasing thermal conductivity and a linear radiation (Robin) boundary condition. For such heat emission problems, we make use of a first-order differential inequality technique to establish conditions on the data sufficient to guarantee that the blow-up of the solutions does occur or does not occur. In addition, the same technique is used to determine a lower bound for the blow-up time when blow-up occurs.

Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence

We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’s method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

On quasilinear parabolic evolution equations in weighted L p -spaces

In this paper we develop a geometric theory for quasilinear parabolic problems in weighted L p -spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) L p -spaces.

Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems

We prove local existence, uniqueness, Holder regularity in space and time, and smooth dependence in Holder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities in the equations as well as in the boundary conditions are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results (Griepentrog in Adv Differ Equ 12:781–840, 1031–1078, 2007) in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.

Directedness of solution set for some quasilinear multi-valued parabolic problems

Recently the authors proved the existence of solutions within a sector of appropriately defined sub-supersolution for a general class of quasilinear parabolic inclusions with Clarke's generalized gradient. The main goal of this article is to show that the solution set contained in this sector is a directed set with respect to the natural partial ordering of functions. Until now the directedness was obtained by assuming a certain additional one-sided growth condition on Clarke's generalized gradient, and it was still an open problem as to whether one can completely drop this condition. Here we provide an affirmative answer, saying that one can indeed drop this additional condition, and only a local Lq -boundedness condition on Clarke's gradient is sufficient. Unlike in the corresponding elliptic case, we are faced with the additional difficulty that the underlying solution space does not possess lattice structure, which considerably complicates the proof of directedness in the parabolic case considered here.

Dimension splitting for quasilinear parabolic equations

In the current paper, we derive a rigorous convergence analysis for a broad range of splitting schemes applied to abstract nonlinear evolution equations, including the Lie and Peaceman-Rachford splittings. The analysis is in particular applicable to (possibly degenerate) quasilinear parabolic problems and their dimension splittings. The abstract framework is based on the theory of maximal dissipative operators, and we both give a summary of the used theory and some extensions of the classical results. The derived convergence results are illustrated by numerical experiments.

Nonlinear parabolic problems in perforated domains

The behavior of the remainder term of the asymptotic expansion for solutions of a quasi-linear parabolic Cauchy-Dirichlet problem in a sequence of domains with fine-granulated boundary is studied. By using a modification of the asymptotic expansion and new pointwise estimates of solutions of the model problem, the uniform convergence of the remainder term to zero is proved.

A quasilinear parabolic singular perturbation problem

We study a singular perturbation problem for a quasilinear uniformly parabolic operator of interest in combustion theory. We obtain uniform estimates, we pass to the limit and we show that, under suitable assumptions, the limit function u is a solution to the free boundary problem {\rm div } F(\nabla u)-\partial_{t}u=0 in \{ u>0 \} , u_\nu=\alpha(\nu, M) on \partial\{ u>0 \} , in a pointwise sense and in a viscosity sense. Here \nu is the inward unit spatial normal to the free boundary \partial\{ u>0 \} and M is a positive constant. Some of the results obtained are new even when the operator under consideration is linear.

On convergence of solutions to equilibria for quasilinear parabolic problems

We show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional C 1 -manifold which is normally hyperbolic. Our results do not depend on the presence of an appropriate Lyapunov functional as in the Łojasiewicz–Simon approach, but are of local nature.

Method of lines solutions of the parabolic inverse problem with an overspecification at a point

The present work is motivated by the desire to obtain numerical solution to a quasilinear parabolic inverse problem. The solution is presented by means of the method of lines. Method of lines is an alternative computational approach which involves making an approximation to the space derivatives and reducing the problem to a system of ordinary differential equations in the variable time, then a proper initial value problem solver can be used to solve this ordinary differential equations system. Some numerical examples and also comparison with finite difference methods will be investigated to confirm the efficiency of this procedure.

Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size

The goal of this paper is to study the asymptotic behavior of the solution of the quasilinear parabolic boundary value problems defined on cylindrical domains when one or several directions go to infinity. We show that the dimension of the space can be reduced and the rate of convergence is analyzed. The evolution p -Laplacian equations and the generalized heat problems are considered.

Implicit difference methods for quasilinear parabolic functional differential problems of the Dirichlet type

Classical solutions of quasilinear functional differential equations are approximated with solutions of implicit difference schemes. Proofs of convergence of the difference methods are based on a comparison technique. Nonlinear estimates of the Perron typ

A two-dimensional metastable flame-front and a degenerate spike-layer problem

A formal asymptotic analysis is used to analyze the metastable behavior associated with a nonlocal PDE model describing the upward propagation of a flame-front interface in a vertical channel with a two-dimensional convex cross-section. In a certain asymptotic limit, the flame-front interface assumes a roughly paraboloidal shape with the tip of the paraboloid drifting asymptotically exponentially slowly towards the closest point on the wall of the channel. Asymptotic estimates for the exponentially small eigenvalues responsible for this metastable behavior are derived together with an explicit ODE for the slow motion of the tip of the paraboloid. The subsequent slow motion of the tip along the channel wall is also characterized explicitly. The analysis is based on a nonlinear transformation that has the effect of transforming the paraboloidal interface to a spike-layer solution of a specific singularly perturbed quasilinear parabolic problem with a nondifferentiable quasilinear term.

The regularity of generalized solutions for the n-dimensional quasi-linear parabolic diffraction problems

In this paper, we study the regularity of generalized solutions u ( x , t ) for the n-dimensional quasi-linear parabolic diffraction problem. By using various estimates and Steklov average methods, we prove that (1): for almost all t the first derivatives u x ( x , t ) are Hölder continuous with respect to x up to the inner boundary, on which the coefficients of the equation are allowed to be discontinuous; and (2): the first derivative u t ( x , t ) is Hölder continuous with respect to ( x , t ) across the inner boundary.