Quasilinear Problems Research Articles

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.

Higher order approximation in exponential form based on half-step grid-points for 2D quasilinear elliptic BVPs on a variant domain

This paper reports a new fourth order Finite Difference Method (FDM) in exponential form for two-dimensional quasilinear boundary value problem of elliptic type (BVPE) with variant solution domain. Further, this discretization is extended to solve the system of quasilinear BVPEs. Following are the main highlights of the proposed FDM:• An unequal mesh 9-point compact stencil is used to approximate the solution. Half-step points are used to evaluate the known variables of this problem. The convergence theory is studied for unequal mesh to validate the fourth order convergence of the suggested FDM.• It is applicable to BVPE irrespective of coordinate systems. Various benchmark problems, for example, Poisson equation in cylindrical coordinates, Burgers’ equation, Navier-Stokes (NS) equations in cylindrical and rectangular coordinates, are solved to depict their fourth order convergence.• Numerical results confirm the accuracy, trustworthiness and acceptability of the suggested numerical algorithm. These results endorse the superiority of the proposed FDM over the previously existing techniques of Mohanty and Kumar (2014), Mohanty and Setia (2014), Priyadarshini and Mohanty (2021).

On the Coupled of NBEM and CFEM for an Anisotropic Quasilinear Problem in an Unbounded Domain with a Concave Angle

On the Coupled of NBEM and CFEM for an Anisotropic Quasilinear Problem in an Unbounded Domain with a Concave Angle

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[ $.

Multiplicity for a strongly singular quasilinear problem via bifurcation theory

A [Formula: see text]-Laplacian elliptic problem in the presence of both strongly singular and [Formula: see text]-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive [Formula: see text]-direction and bifurcates from infinity at [Formula: see text]. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and in particular cases, global multiplicity.

Nonhomogeneous degenerate quasilinear problems with convection

The aim of the paper is to study a Dirichlet problem whose equation is driven by a degenerate p-Laplacian with a weight depending on the solution and whose reaction is a convection term, thus depending on the solution and its gradient. The existence of a weak solution is proven by arguing through a truncated auxiliary problem. A major part of the proof consists in showing that the solutions are bounded.

Transverse instability of high frequency weakly stable quasilinear boundary value problems

This work intends to prove that strong instabilities may appear for high order geometric optics expansions of weakly stable quasilinear hyperbolic boundary value problems, when the forcing boundary term is perturbed by a small amplitude oscillating function, with a transverse frequency. Since the boundary frequencies lie in the locus where the so-called Lopatinskii determinant is zero, the amplifications on the boundary give rise to a highly coupled system of equations for the profiles. A simplified model for this system is solved in an analytical framework using the Cauchy-Kovalevskaya theorem as well as a version of it ensuring analyticity in space and time for the solution. Then it is proven that, through resonances and amplification, a particular configuration for the phases may create an instability, in the sense that the small perturbation of the forcing term on the boundary interferes at the leading order in the asymptotic expansion of the solution. Finally we study the possibility for such a configuration of frequencies to happen for the isentropic Euler equations in space dimension three.

An abstract sub-supersolution method in partially ordered spaces

We present a general framework in which sub-supersolution techniques can be applied. Specifically, we consider a normed vector space endowed with a partial order. This allows us to introduce a notion of sub- and supersolutions relative to an equation driven by operators defined from the space to its topological dual. We provide sufficient conditions to guarantee that the interval determined by an ordered pair of sub-supersolutions contains a solution of the initial problem. We also study existence of extremal positive or negative solutions and location of nodal solutions. Examples of applications to quasilinear boundary value problems are given.

Multiple solutions for a class of quasilinear problems with double criticality

AbstractWe establish multiplicity results for the following class of quasilinear problems P\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*}where $\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$ for a generalized N-function $\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$. We consider $\Omega \subset \mathbb {R}^{N}$ to be a smooth bounded domain that contains two disjoint open regions $\Omega _N$ and $\Omega _p$ such that $\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$. The main feature of the problem $(P)$ is that the operator $-\Delta _{\Phi }$ behaves like $-\Delta _N$ on $\Omega _N$ and $-\Delta _p$ on $\Omega _p$. We assume the nonlinearity $f:\Omega \times \mathbb {R}\to \mathbb {R}$ of two different types, but both behave like $e^{\alpha |t|^{\frac {N}{N-1}}}$ on $\Omega _N$ and $|t|^{p^{*}-2}t$ on $\Omega _p$ as $|t|$ is large enough, for some $\alpha >0$ and $p^{*}=\frac {Np}{N-p}$ being the critical Sobolev exponent for $1< p< N$. In this context, for one type of nonlinearity $f$, we provide a multiplicity of solutions in a general smooth bounded domain and for another type of nonlinearity $f$, in an annular domain $\Omega$, we establish existence of multiple solutions for the problem $(P)$ that are non-radial and rotationally non-equivalent.

Lower bounds for eigenvalues of even ordered quasilinear differential equations

In this paper, we establish lower bounds for the eigenvalues of even ordered quasilinear eigenvalue problems with indefinite weights. We first consider the case when the weight function is identically one and later generalize the results to any weight function within the p p -th Muckenhopt class. In the process, we obtain new Lyapunov-type inequalities for the corresponding problems and utilize them to obtain the lower bound of the k k -th eigenvalues. In particular, we show that the first eigenvalue is bounded below in terms of the integral of the weight instead of the integral of its positive part. Our results provide a much sharper bound than existing results.

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.

Existence of heteroclinic and saddle-type solutions for a class of quasilinear problems in whole ℝ2

In this work, we use variational methods to prove the existence of heteroclinic and saddle-type solutions for a class quasilinear elliptic equations of the form [Formula: see text] where [Formula: see text] is a N-function, [Formula: see text] is a periodic positive function and [Formula: see text] is modeled on the Ginzburg–Landau potential. In particular, our main result includes the case of the potential [Formula: see text], which reduces to the classical double well Ginzburg–Landau potential when [Formula: see text], that is, when we are working with the Laplacian operator.

Results of Semigroup of Linear Operator Generating a Quasilinear Equations of Evolution

In this paper, results of $\omega$-order preserving partial contraction mapping generating a quasilinear equation of evolution were presented. In general, the study of quasilinear initial value problems is quite complicated. For the sake of simplicity we restricted this study to the mild solution of the initial value problem of a quasilinear equation of evolution. We show that if the problem has a unique mild solution $v\in C([0,T]: X)$ for every given $u\in C([0,T]:X)$, then it defines a mapping $u\to v=F(u)$ of $C([0,T]:X)$ into itself. We also show that under the suitable condition, there exists always a $T',\ 0<T'\leq T$ such that the restriction of the mapping $F$ to $C([0,T']:X)$ is a contraction which maps some ball of $C([0,t']:X)$ into itself by proving the existence of a local mild solution of the initial value problem.

Multiplicity of Solutions for Fractional-Order Differential Equations via the κ(x)-Laplacian Operator and the Genus Theory

In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χ-fractional space Hκ(x)γ,β;χ(Δ). Using the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem, we show that under certain suitable assumptions the considered problem has at least k pairs of non-trivial solutions.

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.

Asymptotics and Confluence for a Singular Nonlinearq-Difference-Differential Cauchy Problem

We examine a family of nonlinearq−difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilationq−difference operators, recently investigated by the author, and quasilinear Kowalevski type problems that involve contractionq−difference operators. We build up local holomorphic solutions to these problems. Two aspects of these solutions are explored. One facet deals with asymptotic expansions in the complex time variable for which a mixed type Gevrey andq−Gevrey structure are exhibited. The other feature concerns the problem of confluence of these solutions asq>1tends to 1.

Existence of solutions for a quasilinear problem with fast nonlocal terms

In this paper, it is considered the existence of a solution for a nonlocal equation involving the p-Laplacian operator which is related to several applications such as vibration problems, flow of fluids through a homogeneous isotropic rigid porous medium, nuclear reactor's dynamics and in population dynamics. The approach is based on sub-supersolution arguments and Schauder's fixed point theorem. An important fact is that our result is valid for nonlocal terms which may exhibit a growth higher than p.

Positive solutions to discontinuous quasilinear problems with critical exponents

Positive solutions to discontinuous quasilinear problems with critical exponents

Local well-posedness for quasi-linear problems: A primer

Proving local well-posedness for quasi-linear problems in partial differential equations presents a number of difficulties, some of which are universal and others of which are more problem specific. On one hand, a common standard for what well-posedness should mean has existed for a long time, going back to Hadamard. On the other hand, in terms of getting there, there are by now both many variations—and also many misconceptions. The aim of these expository notes is to collect a number of both classical and more recent ideas in this direction, and to assemble them into a cohesive roadmap that can be then adapted to the reader’s problem of choice.

Multiplicity of solutions for fractional p-Laplacian equation with indefinite nonlinearity

ABSTRACT We study the existence and multiplicity of positive solutions for the quasi-linear problem where Ω is a bounded domain in , λ a real parameter, 1<p<q and a changes sign.