Quasilinear Parabolic Problems Research Articles

An efficient acceleration technique of implicit schemes for quasi-linear parabolic problems

An efficient acceleration technique of implicit schemes for quasi-linear parabolic problems

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Blow-Up of Solution and Energy Decay for a Quasilinear Parabolic Problem

Existence and Uniqueness Solutions for Some Strongly Quasilinear Parabolic Problems in Anisotropic Sobolev Spaces

Existence and Uniqueness Solutions for Some Strongly Quasilinear Parabolic Problems in Anisotropic Sobolev Spaces

Unconditionally optimal error estimates of linearized Crank-Nicolson virtual element methods for quasilinear parabolic problems on general polygonal meshes

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme.

Uniqueness of a Generalized Solution for a One-Dimensional Thermal Explosion Model of a Compressible Micropolar Real Gas

In this paper, we analyze a quasi-linear parabolic initial-boundary problem describing the thermal explosion of a compressible micropolar real gas in one spatial dimension. The model contains five variables, mass density, velocity, microrotation, temperature, and the mass fraction of unburned fuel, while the associated problem contains homogeneous boundary conditions. The aim of this work is to prove the uniqueness theorem of the generalized solution for the mentioned initial-boundary problem. The uniqueness of the solution, together with the proven existence of the solution, makes the described initial-boundary problem theoretically consistent, which provides a basis for the development of numerical methods and the engineering application of the model.

Analytical and Numerical Investigation of Two-Dimensional Heat Transfer with Periodic Boundary Conditions

A two-dimensional heat diffusion problem with a heat source that is a quasilinear parabolic problem is examined analytically and numerically. Periodic boundary conditions are employed. As the problem is nonlinear, Picard’s successive approximation theorem is utilized. We demonstrate the existence, uniqueness, and constant dependence of the solution on the data using the generalized Fourier method under specific conditions of natural regularity and consistency imposed on the input data. For the numerical solution, an implicit finite difference scheme is used. The results obtained from the analytical and numerical solutions closely match each other.

Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem

Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem

Stability of Solution of Quasilinear Parabolic Two-Dimensional with Inverse Coefficient by Fourier Method

In this article, the heat inverse two-dimensional quasilinear parabolic problem is examined. The stability and numerical solution for the problem are discussed.Since the problem is not linear, Picard's successive approximations theorem is used. In the numerical part, the solution is made with the finite difference and linearization method.

Weak and classical solutions to an asymptotic model for atmospheric flows

In this paper we study a recently derived mathematical model for nonlinear propagation of waves in the atmosphere, for which we establish the local well-posedness in the setting of classical solutions. This is achieved by formulating the model as a quasilinear parabolic evolution problem in an appropriate functional analytic framework and by using abstract theory for such problems. Moreover, for L2-initial data, we construct global weak solutions by employing a two-step approximation strategy based on a Galerkin scheme, where an equivalent formulation of the problem in terms of a new variable is used. Compared to the original model, the latter has the advantage that the L2-norm is a Liapunov functional.

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

<p style='text-indent:20px;'>In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula> by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem <inline-formula><tex-math id="M2">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula>. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.</p>

Quasilinear parabolic Venttsel problem with discontinuous principal coefficients

Получены новые результаты о сильной разрешимости в пространствах Соболева квазилинейной задачи Вентцеля для параболических уравнений с разрывными старшими коэффициентами.

Weak Solutions for a Quasilinear Elliptic and Parabolic Problems Involving the (p(x), q(x))-Laplacian Operator

Weak Solutions for a Quasilinear Elliptic and Parabolic Problems Involving the (p(x), q(x))-Laplacian Operator

The existence of renormalized solution for quasilinear parabolic problem with variable exponents and measure data

In this paper, the study of the existence of a renormalized solution for quasilinear parabolicproblem with variable exponents and measure data. The model is: u_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda\left\vert u\right\vert ^{p(x)-2}u=\mu\text{ } &\text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[,\\u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\u(.,0)=u_{0}(.) & \text{in}\hspace{0.5cm}\Omega, where $ \lambda>0$ and $ T $ is any positive constant, $ \mu\in\mathcal{M}_{0}(Q) $ is any measure with bounded variation over $ Q=\Omega \times ]0,T[ $.

Numerical Solutions of Quasilinear Parabolic Problems by a Continuous Space-Time Finite Element Scheme

In this paper, continuous space-time finite element methods are developed for approximating a class of quasilinear parabolic problems in space and in time, simultaneously. The basis of the whole approach is on a space-time variational formulation where streamline upwind terms and interface jump terms are further added for stabilizing the discretization in the direction of time. Error estimates are shown and are verified numerically through a series of numerical tests. Emphasis is placed on investigating the asymptotic convergence of the error parts, which are related to the time discretization.

Shifted Legendre reproducing kernel Galerkin method for the quasilinear degenerate parabolic problem

This paper focuses on solving a class of quasilinear degenerate parabolic problems by using shifted Legendre reproducing kernel Galerkin method (SLRKGM for short). The quasilinear term was linearized and the time derivative was discretized by using the finite difference scheme. The basis functions were constructed by the shifted Legendre polynomials and the approximate solution was obtained by the Galerkin method. We also discussed the error estimates and stability analysis of the method. Numerical examples demonstrate the feasibility and reliability of our algorithm.

Partial existence result for homogeneous quasilinear parabolic problems beyond the duality pairing

In this paper, we study the existence of distributional solutions solving (1.3) on a bounded domain \(\Omega \) satisfying a uniform capacity density condition where the nonlinear structure \(\mathcal {A}(x,t,\nabla u)\) is modelled after the standard parabolic p-Laplace operator. In this regard, we need to prove a priori estimates for the gradient of the solution below the natural exponent and a higher integrability result for very weak solutions at the initial boundary. The elliptic counterpart to these two estimates is fairly well developed over the past few decades, but no analogous theory exists in the quasilinear parabolic setting. Two important features of the estimates proved here are that they are non-perturbative in nature and we are able to take non-zero boundary data. As a consequence, our a priori estimates are new even for the heat equation on bounded domains. This partial existence result is a nontrivial extension of the existence theory of very weak solutions from the elliptic setting to the quasilinear parabolic setting. Even though we only prove partial existence result, nevertheless we establish the necessary framework that when proved would lead to obtaining the full result for the homogeneous problem.

Translating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz–Minkowski plane R12

In this paper, we investigate the evolution of spacelike curves in the Lorentz–Minkowski plane R12 along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike grim reaper curve as time tends to infinity.

Nonexistence of solutions to quasilinear parabolic equations with a potential in bounded domains

We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in \(\Omega \times (0,+\infty )\), where \(\Omega \) is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.

Stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem

In this work, we present results on stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem. We show that this nonlocal version of the well-known Chafee-Infante equation bears some resemblance with the local version. However, its nonlocal characteristic requires a fine analysis of the spectrum of the associated linear operators, a lot more elaborated than the local case. The saddle point property of equilibria is shown to hold for this quasilinear model.

On Qualitative Properties of Sign-Constant Solutions of Some Quasilinear Parabolic Problems

We study the Cauchy problem for quasilinear parabolic inequalities containing squares of the first derivatives of an unknown function (the so-called nonlinearities of the KPZ type). The coefficients of the leading nonlinear terms of the inequalities considered either can be continuous functions (the regular case) or can admit power singularities (the singular case) of degree no greater than 1. For the regular case, we prove the damping of global nonnegative solutions to the problem studied. By damping, we mean the boundedness of the support of a solution for each positive t, uniform (with respect to t) convergence to zero as |x| → ∞, and vanishing (for any x) starting with a certain sufficiently large t. For the singular case, we proved that the problem considered has no global positive solutions.