Quasilinear Parabolic Problems Research Articles

A quasilinear parabolic problem with a source term and a nonlocal absorption

We investigate a quasi-linear parabolicproblem with nonlocal absorption, for which the comparison principle is not always available. Thesufficient conditions are established via energy method to guaranteesolution to blow up or not, and the long time behavior is alsocharacterized for global solutions.

Liouvilie Type Results for a Class of Quasilinear Parabolic Problem

In this paper, we investigate the parabolic logistic equation with blow-up initial and boundary values on a smooth bounded domain where T>0. Under suitable assumptions on a(x, t) and f, we show that such solution stays bounded in any compact subset of Ω as t increase to T. Other asymptotic estimates will given in this work.

Rate of convergence for one-dimensional quasilinear parabolic problem and its applications

Based on a comparison principle, we derive an exponential rate of convergence for solutions to the initial–boundary value problem for a class of quasilinear parabolic equations in one space dimension. We then apply the result to some models in population dynamics and image processing.

Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations

This paper studies the Sobolev regularity for weak solutions of a class of singular quasi-linear parabolic problems of the form ut−div[A(x,t,u,∇u)]=div[F] with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients A are discontinuous and singular in (x,t)-variables, and dependent on the solution u. Global and interior weighted W1,p(ΩT,ω)-regularity estimates are established for weak solutions of these equations, where ω is a weight function in some Muckenhoupt class of weights. The results obtained are even new for linear equations, and for ω=1, because of the singularity of the coefficients in (x,t)-variables.

Existence of periodic solutions for some quasilinear parabolic problems with variable exponents

In this paper, we prove the existence of at least one periodic solution for some nonlinear parabolic boundary value problems associated with Leray–Lions’s operators with variable exponents under the hypothesis of existence of well-ordered sub- and supersolutions.

Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions

We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local invariant manifolds near a given equilibrium. In a companion paper, we treat center, center-stable and center-unstable manifolds for such problems and investigate their stability properties. This theory applies e.g. to reaction-diffusion systems with dynamical boundary conditions and to the two-phase Stefan problem with surface tension.

The lower bound for the blowup time of the solution to a quasi-linear parabolic problem

In this paper, using a delicate application of general Sobolev inequality, we establish the lower bound for the blowup time of the solution to a quasi-linear parabolic problem, which improves the result of Theorem 2.1 in Bao and Song (2014).

A survey on second order free boundary value problems modelling MEMS with general permittivity profile

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles. The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

Well-posedness of a quasilinear evolution problem modelling MEMS with general permittivity

Subject of consideration is the analytical investigation of a coupled system of partial differential equations arising from the modelling of electrostatically actuated microelectromechanical systems with general permittivity profile. A quasilinear parabolic evolution problem for the displacement u of an elastic membrane is coupled with an elliptic free boundary value problem that determines the electrostatic potential \(\psi \) in the region between the elastic membrane and a rigid ground plate. The system is shown to be well-posed locally in time for all arbitrarily large values \(\lambda \) of the applied voltage, whereas small values of the applied voltage, which do not exceed a certain critical value \(\lambda _{*}\), do even allow globally in time existing solutions. In addition, conditions are specified which force solutions emerging from a non-positive initial deflection to stay non-positive as long as they exist.

A note on weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation of generalized polytropic filtration

In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin, as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under Lipschitzity of the nonlinear external forces, [Formula: see text] and [Formula: see text], we obtain the uniqueness of the weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result we prove the existence of the unique strong probabilistic solution.

Averaging of Parabolic Initial-Boundary Value Problems with Rapidly Oscillating Principal Part

We justify the Krylov–Bogolyubov averaging method for linear and quasilinear parabolic second order initial-boundary value problems with rapidly time-oscillating principal terms.

Global a priori bounds for weak solutions to quasilinear parabolic equations with nonstandard growth

In this paper we study a rather wide class of quasilinear parabolic problems with nonlinear boundary condition and nonstandard growth terms. It includes the important case of equations with a p(t,x)-Laplacian. By means of the localization method and De Giorgi’s iteration technique we derive global a priori bounds for weak solutions of such problems. Our results seem to be new even in the constant exponent case.

Weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation with nonstandard growth

In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in [45] as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.

Quasilinear parabolic problem with p(x)-laplacian: existence, uniqueness of weak solutions and stabilization

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $$ \left\{ \begin{array}{ll} u_t-\Delta _{p(x)}u = f(x,u)&\quad \text{in } \quad Q_T \stackrel{{\rm{def}}}{=} (0,T)\times\Omega, u = 0 & \quad\text{on} \quad \Sigma_T\stackrel{{\rm{def}}}{=} (0,T)\times\partial\Omega, u(0,x)=u_0(x)& \quad \text{in} \quad \Omega \end{array} \right. \quad\quad (P_{T}) $$ involving the p(x)-laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.

Classical solutions to quasilinear parabolic problems with dynamic boundary conditions

We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.

New patterns of travelling waves in the generalized Fisher–Kolmogorov equation

We prove the existence and uniqueness of a family of travelling waves in a degenerate (or singular) quasilinear parabolic problem that may be regarded as a generalization of the semilinear Fisher–Kolmogorov–Petrovski–Piscounov equation for the advance of advantageous genes in biology. Depending on the relation between the nonlinear diffusion and the nonsmooth reaction function, which we quantify precisely, we investigate the shape and asymptotic properties of travelling waves. Our method is based on comparison results for semilinear ODEs.

Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds

We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.

Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces

Convergence results are shown for full discretizations of quasilinear parabolic partial differential equations on evolving surfaces. As a semidiscretization in space the evolving surface finite element method is considered, using a regularity result of a generalized Ritz map, optimal order error estimates for the spatial discretization is shown. Combining this with the stability results for Runge–Kutta and backward differentiation formulae time integrators, we obtain convergence results for the fully discrete problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1200–1231, 2016

Finite-Time Singularities of Solutions to Microelectromechanical Systems with General Permittivity

Of concern is a study of qualitative properties of solutions to the evolution problem modelling microelectromechanical systems with general permittivity. The system couples a quasilinear parabolic evolution problem for a membrane’s displacement with an elliptic free boundary value problem for the electrostatic potential in the region between the membrane and a rigid ground plate. It is shown that, under a structural condition on the permittivity profile, non-positive solutions develop a singularity in finite time, provided that the applied voltage is large enough and the aspect ratio of the system is small enough.

Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

Abstract In this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.