Quasilinear Parabolic Equations Research Articles

Well-posedness of quasilinear parabolic equations in time-weighted spaces

Abstract Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the nonlinearities $u\mapsto f(u)$ and $u\mapsto A(u)$ . Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.

Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group

In the Heisenberg group Hn, we obtain the local second-order HWloc2,2-regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ∂tu=∑i=12nXiAi(Xu) modeled on the parabolic p-Laplacian equation. Specifically, when 2≤p≤4, we demonstrate the integrability of (∂tu)2, namely, ∂tu∈Lloc2; when 2≤p<3, we demonstrate the HWloc2,2-regularity of u, namely, XXu∈Lloc2. For the HWloc2,2-regularity, when p≥2, the range of p is optimal compared to the Euclidean case.

A Thin Film Model for Meniscus Evolution

In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of liquid, i.e. a liquid contained between two solid surfaces and part of the liquid surface is in contact with the air. The liquid is governed by Navier–Stokes equation, while the contact point, i.e. where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle. We consider a scaling limit of the liquid thickness (lubrication approximation) and the contact angle between the liquid–solid and the liquid–gas interfaces close to π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}. This resulting model is a free boundary problem for the equation ht+(h3hxxx)x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_t + (h^3h_{xxx})_x = 0$$\\end{document}, for which we have h>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h>0$$\\end{document} at the contact point (different from the usual thin film equation with h=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h=0$$\\end{document} at the contact point). We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed. Furthermore, the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from the no-slip condition. Additionally, we show the global stability of steady state solutions in a periodic setting.

Parabolic Finslerian Allen–Cahn equation and its Harnack inequality

Parabolic Finslerian Allen–Cahn equation is a quasilinear parabolic equation on Finsler metric measure spaces. In this paper, we establish a Harnack inequality for solutions to this equation on compact Finsler metric measure spaces using geometric methods. Moreover, we give a degree 4 polynomial gradient estimate of the traveling wave solution on flat-curved spaces.

Finite-time property of a mechanical viscoelastic system with nonlinear boundary conditions on corner-Sobolev spaces

In this article, we deal with the initial boundary value problem for a viscoelastic system related to the quasilinear parabolic equation with nonlinear boundary source term on a manifold $\mathbb{M}$ with corner singularities. We prove that, under certain conditions on relaxation function $g$, any solution $u$ in the corner-Sobolev space $\mathcal{H}^{1,(\frac{N-1}{2},\frac{N}{2})}_{\partial^{0}\mathbb{M}}(\mathbb{M})$ blows up in finite time. The estimates of the life-span of solutions are also given.

Two-level implicit high-order compact scheme in exponential form for 3D quasi-linear parabolic equations

We discuss a new high accuracy compact exponential scheme of order four in space and two in time to solve the three-dimensional quasi-linear parabolic partial differential equations. The derived half-step discretization based scheme is implicit in nature and demands only two levels for computation. The generalization of the proposed exponential scheme for the system of the quasi-linear parabolic PDEs is also represented. We generate unconditionally stable alternating direction implicit scheme for the linear parabolic equation in general form. The accuracy and the theoretical results of the proposed scheme are verified for high Reynolds number by several numerical problems like linear and non-linear convection-diffusion equation, coupled Burgers' equations, Navier-Stokes equations, quasi-linear parabolic equation, etc.

On the Nonstandard Maximum Principle and Its Application for Construction of Monotone Finite-Difference Schemes for Multidimensional Quasilinear Parabolic Equations

On the Nonstandard Maximum Principle and Its Application for Construction of Monotone Finite-Difference Schemes for Multidimensional Quasilinear Parabolic Equations

Local behavior for solutions to anisotropic weighted quasilinear degenerate parabolic equations

Local behavior for solutions to anisotropic weighted quasilinear degenerate parabolic equations

Semilinear approximations of quasilinear parabolic equations with applications

An approach to solvability of certain quasilinear parabolic equations is presented by approximating the quasilinear equation under consideration with a parameter family of semilinear problems with stronger linear fractional diffusion term. Defined on arbitrarily long time intervals, solutions to the original problem are found as a suitable limit of global solutions to those semilinear approximations. The method is applied to nonlinear parabolic Kirchhoff equation, quasilinear reaction–diffusion equation, and critical 2D surface quasi‐geostrophic equation associated with Dirichlet boundary conditions.

Global and blow-up results for a quasilinear parabolic equation with variable sources and memory terms

Global and blow-up results for a quasilinear parabolic equation with variable sources and memory terms

Group classification for one type of space-time fractional quasilinear parabolic equation

Group classification for one type of space-time fractional quasilinear parabolic equation

On a quasilinear parabolic–hyperbolic system arising in MEMS modeling

AbstractA coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Holder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

Nonlinear random perturbations of PDEs and quasi-linear equations in Hilbert spaces depending on a small parameter

We study a class of quasi-linear parabolic equations defined on a separable Hilbert space, depending on a small parameter in front of the second order term. Through the nonlinear semigroup associated with such equation, we introduce the corresponding SPDE and we study the asymptotic behavior of its solutions, depending on the small parameter. We show that a large deviations principle holds and we give an explicit description of the action functional.

Unbounded Sturm attractors for quasilinear parabolic equations

AbstractWe analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.

Curve shortening flow on Riemann surfaces with conical singularities

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some contracting and convergence results.

On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L 2 -norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

A survey of results of St.Petersburg State University research school on nonlinear partial differential equations I

The article contains a review of the most important results obtained in the framework of the St.Petersburg State University research school on nonlinear PDEs (the O.A. Ladyzhenskaya - N.N. Uraltseva school). The main attention is paid to the works carried out at our university over the past 50 years. The first part of the review concerns the solvability and qualitative properties of solutions to boundary value problems for the second order scalar quasilinear elliptic and parabolic equations, as well as variational problems. The planned second part of the review will include sections on fully nonlinear equations and systems of equations and on free boundary problems.

ON THE EXISTENCE OF STRONG SOLUTIONS OF HIGHER ORDER QUASILINEAR PARABOLIC EQUATIONS

We consider boundary value problems for quasilinear parabolic equations with a main quasilinear elliptic operator of order 2b ≥ 2 in Sobolev space W2b,1 p (QT ).

Qualitative Analysis of an Optimal Sparse Control Problem for Quasi-Linear Parabolic Equation with Variable Order of Nonlinearity

Qualitative Analysis of an Optimal Sparse Control Problem for Quasi-Linear Parabolic Equation with Variable Order of Nonlinearity