Quasilinear Parabolic Equations Research Articles

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

AbstractWe 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.

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.

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.

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

Wellposedness of a nonlinear parabolic-dispersive coupled system modelling MEMS

In this paper we study the local wellposedness of the solution to a non-linear parabolic-dispersive coupled system which models a Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device has two parallel plates separated by a gas-filled thin gap. The nonlinear parabolic-dispersive coupled system modelling the device consists of a quasilinear parabolic equation for the gas pressure and a semilinear plate equation for gap width. We show the local-in-time existence of strict solutions for the system, by combining a local-in-time existence result for the dispersive equation, Hölder continuous dependence of its solution on that of the parabolic equation, and then local-in-time existence for a resulting abstract parabolic problem. Semigroup approaches are vital for both main parts of the problem.

Optimal Sparse Control Formulation for Reconstruction of Noise-Affected Images

We discuss the optimal control formulation for enhancement and denoising of satellite multiband images and propose to take it in the form of an L1 control problem for a quasi-linear parabolic equation with a nonlocal p[u] Laplacian and with a cost functional of a tracking type. The main characteristic features of the considered parabolic problem is that the variable exponent p(t,x) and the diffusion anisotropic tensor D(t,x) are not predefined well a priori; instead, these characteristics nonlocally depend on the form of the solution of this problem (i.e., pu=p(t,x,u) and Du=D(t,x,u)). We prove the existence of optimal pairs with sparse L1 controls used for the indirect approach and a special family of approximation problems.

Potential Estimates and Quasilinear Parabolic Equations with Measure Data

In this memoir, we study the existence and regularity of the quasilinear parabolic equations: u t − div ⁡ ( A ( x , t , ∇ u ) ) = B ( u , ∇ u ) + μ , \begin{equation*} u_t-\operatorname {div}(A(x,t,\nabla u))=B(u,\nabla u)+\mu , \end{equation*} in either R N + 1 \mathbb {R}^{N+1} or R N × ( 0 , ∞ ) \mathbb {R}^N\times (0,\infty ) or on a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 \Omega \times (0,T)\subset \mathbb {R}^{N+1} where N ≥ 2 N\geq 2 . We shall assume that the nonlinearity A A fulfills standard growth conditions, the function B B is a continuous and μ \mu is a radon measure. Our first task is to establish the existence results with B ( u , ∇ u ) = ± | u | q − 1 u B(u,\nabla u)=\pm |u|^{q-1}u , for q > 1 q>1 . We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with B ≡ 0 B\equiv 0 , under minimal conditions on the boundary of domain and on nonlinearity A A . Finally, due to these estimates, we solve the existence problems with B ( u , ∇ u ) = | ∇ u | q B(u,\nabla u)=|\nabla u|^q for q > 1 q>1 .

Linear and quasilinear evolution equations in the context of weighted Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}-spaces

In 2004, the article Maximal regularity for evolution equations in weightedLp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}-spaces by Prüss and Simonett (Arch Math 82:415–431, 2004) has been published in Archiv der Mathematik. We provide a survey of the main results of that article and outline some applications to semilinear and quasilinear parabolic evolution equations which illustrate their power.

Upper estimates for blow-up solutions of a quasi-linear parabolic equation

In this paper, we consider a quasi-linear parabolic equation u_t=u^p(x_{xx}+u). It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for p>2. We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of 2<p<3 and in the case of p=3, separately. In addition, we assert that if 2 le p le 3 then lim _{t nearrow T}(T-t)^{frac{1}{p}+varepsilon }max u(x,t)=0 z for any varepsilon >0 under some assumptions.

Existence and Uniqueness of the Initial-Boundary Value Problem Solution of Multicomponent Sediments Transport in Coastal Marine Systems

Introduction. This work is devoted to the study of a non-stationary two-dimensional model of sediment transport in coastal marine systems. The model takes into account the complex multi-fractional composition of sediments, the gravity effect and tangential stress caused by the impact of waves, turbulent exchange, dynamically changing bottom topography, and other factors. The aim of the work was to carry out an analytical study of the conditions for the initialboundary value problem existence and uniqueness corresponding to the specified model.Materials and Methods. Linearization of the initial-boundary value problem is performed on a temporary uniform grid. The nonlinear coefficients of a quasilinear parabolic equation are taken with a “delay” by one grid step. Thus, a chain of correlated by initial conditions is the final solutions of problems is built. The study of the existence and uniqueness of the problems included in this chain, and therefore the original problem as a whole, is carried out involving the methods of mathematical and functional analysis, as well as methods for solving differential equations.Results. Earlier, the authors investigated the existence and uniqueness of the initial-boundary value problem of the transport of sediments of a single-component composition. In the present work, the result obtained is extended to the case of multi-fractional sediments.Discussion and Conclusions. The non-linear spatial two-dimensional model of sediment transport was previously investigated by the team of authors in the case of bottom sediments consisting of particles having the same characteristic dimensions and density (single-component composition) based on the analysis of the existing results of mathematical modeling of hydrodynamic processes. In this paper, the previous results of the study are extended to the case of sediments of a multicomponent composition, namely, the conditions for the existence and uniqueness of the solution of the initial-boundary value problem corresponding to the considered model are determined.

A continuum model for morphology formation from interacting ternary mixtures: Simulation study of the formation and growth of patterns

Our interest lies in exploring the ability of a coupled nonlocal system of two quasilinear parabolic partial differential equations to produce phase separation patterns. The obtained patterns are referred here as morphologies. Our target system is derived in the literature as the rigorous hydrodynamic limit of a suitably scaled interacting particle system of Blume–Capel-type driven by Kawasaki dynamics. The system describes in a rather implicit way the interaction within a ternary mixture that is the macroscopic counterpart of a mix of two populations of interacting solutes in the presence of a solvent. Our discussion is based on the qualitative behavior of numerical simulations of finite volume approximations of smooth solutions to our system and their quantitative postprocessing in terms of two indicators (correlation and structure factor calculations). Our results show many similar qualitative features (e.g. general shape and approximate coarsening rates) which have been observed in previous works on the stochastic Blume–Capel model with three interacting species. The properties of the obtained morphologies (shape, connectivity, and so on) can play a key role in, e.g., the design of the active layer for efficient organic solar cells.

Gradient estimates of the solution to the Cauchy problem for degenerate parabolic equations with nonpower nonlinearities

ABSTRACT We prove the uniform gradient estimates of the nonnegative solution to the Cauchy problem for quasilinear degenerate parabolic equations with nonpower nonlinearity. The large-time estimates in terms of initial data mass and nonlinearity structure were obtained. In the particular case, when the nonlinearity of the equation is power like, then our result coincides with well-known optimal bound.

Two Free Boundaries Problem for a Parabolic Equation

В данной работе рассматривается задача типа Стефана с двумя свободными границами для квазилинейного параболического уравнения в одномерном случае. Исследование нелинейных задач со свободными границами методом, основанным на построении априорных оценок. Поэтому сначала устанавливаются некоторые первоначальные априорные оценки для решения рассматриваемой задачи. Основной трудностью при построении теории для задач квазилинейных параболических уравнений второго порядка является получение априорной оценки модуля производной решение, а также в задачах со свободной границей требуются дополнительные рассуждения. Для этого задача сводится к задаче с фиксированной границей через замену переменных. Полученная задача имеет зависящие от времени и положения в пространстве коэффициенты с нелинейными слагаемыми. Далее построены априорных оценок типа Шаудера для решения уравнения с нелинейными слагаемыми и закрепленной границей. На основе полученных оценок доказана единственность решения задачи. Затем мы доказываем глобальное существование решения задачи с помощью теоремы Лерэ-Шаудера о неподвижной точке. This paper considers a two-free-boundary Stefan-type problem for a quasi-linear parabolic equation in one dimension. Nonlinear problems with free boundaries are studied using a method based on constructing a priori estimates. Therefore, some initial a priori estimates for the solution to the problem under consideration are first established. The main difficulty in constructing a theory for second-order quasi-linear parabolic equations is obtaining an a priori estimate for the solution’s derivative module, and additional arguments are required in problems with a free boundary. To address this, the problem is reduced to a fixed-boundary problem through a change of variables. The resulting problem has time- and spacedependent coefficients with nonlinear terms. Next, Schauder-type a priori estimates are constructed for the equation with nonlinear terms and a fixed boundary. Based on these estimates, the uniqueness of the solution to the problem is proven. Then, the global existence of the solution to the problem is demonstrated using the Leray-Schauder fixed-point theorem.

Strong periodic solutions to quasilinear parabolic equations: An approach by the Da Prato–Grisvard theorem

AbstractThis article develops an approach to unique, strong periodic solutions to quasilinear evolution equations by means of the classical Da Prato–Grisvard theorem on maximal ‐regularity in real interpolation spaces. The method is used to show that quasilinear Keller–Segel systems admit a unique, strong ‐periodic solution in a neighborhood of 0 provided the external forces are ‐periodic and satisfy certain smallness conditions. A similar assertion applies to a Nernst–Planck–Poisson type system in electrochemistry. The proof for the quasilinear Keller–Segel systems relies also on a new mixed derivative theorem in real interpolation spaces, that is, Besov spaces, which is of independent interest.

Besov estimates for weak solutions of a class of quasilinear parabolic equations with general growth

AbstractIn this paper, we obtain the local higher fractional differentiability regularity estimates in Besov spaces of weak solutions for the quasilinear parabolic equations with general growth under some proper conditions on , and . Moreover, we would like to point out that the results in the present work improve the known results for such parabolic equations.