Higher-order Parabolic Equation Research Articles

Investigating invariance and conservation laws for the classes of nonlinear parabolic and wave systems

We study various classes of the nonlinear dynamics of some high order parabolic equations like the Oskolkov–Benjamin–Bona–Mahony–Burgers and Benjamin–Bona–Mahony–Peregrine–Burger equations that arise in the study of some wave phenomena. Also, a broader class of partial differential equations are used in modelling ocean waves that originate from the Ostrovsky equation. We study the invariance properties via the Lie invariance method for the nonlinear systems and further establish classes of conservation laws for models arises in this study. We show how the relationship leads to double reductions of the systems. This relationship is determined by a recent result involving multipliers that lead to a total divergence or closed form of the differential equation under investigation.

A solution to the Ricquier problem for a high-order parabolic equation in an unbounded domain: questions of existence and stabilization

The existence of a solution to the Ricquier problem with Dirichlet boundary conditions for one high-order parabolic equation in an unbounded domain is discussed. The convergence of the approximation using the Galerkin method of a generalized solution to the Ricquier problem is investigated. The dependence of behavior of the norm of the problem solution for large values of time on the geometry of an area unlimited in spatial variables lying at the base of the cylinder with a limited carrier of the initial function is considered. For a fairly wide class of unbounded domains, upper bounds for the solution of a mixed problem for a high-order parabolic equation are estimated, which are expressed in terms of simple geometric characteristics of the unbounded domain. The obtained upper estimations for the solution of the mixed problem are accurate.

Efficient numerical approaches with accelerated graphics processing unit (GPU) computations for Poisson problems and Cahn-Hilliard equations.

In this computational paper, we focused on the efficient numerical implementation of semi-implicit methods for models in materials science. In particular, we were interested in a class of nonlinear higher-order parabolic partial differential equations. The Cahn-Hilliard (CH) equation was chosen as a benchmark problem for our proposed methods. We first considered the Cahn-Hilliard equation with a convexity-splitting (CS) approach coupled with a backward Euler approximation of the time derivative and tested the performance against the bi-harmonic-modified (BHM) approach in terms of accuracy, order of convergence, and computation time. Higher-order time-stepping techniques that allow for the methods to increase their accuracy and order of convergence were then introduced. The proposed schemes in this paper were found to be very efficient for 2D computations. Computed dynamics in 2D and 3D are presented to demonstrate the energy-decreasing property and overall performance of the methods for longer simulation runs with a variety of initial conditions. In addition, we also present a simple yet powerful way to accelerate the computations by using MATLAB built-in commands to perform GPU implementations of the schemes. We show that it is possible to accelerate computations for the CH equation in 3D by a factor of 80, provided the hardware is capable enough.

$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients

$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients

Homogenization of the Higher-Order Parabolic Equations with Periodic Coefficients

Homogenization of the Higher-Order Parabolic Equations with Periodic Coefficients

Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order

We study the long-time behavior of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model, for instance, the dynamic behavior of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power law or Ellis law for the fluid viscosity. In all three cases, positive constants (i.e., positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behavior of solutions with respect to the -norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state converge to equilibrium at rate for some . Finally, in the case of an Ellis fluid, steady states are exponentially stable in .

Mesh Optimization for the Acoustic Parabolic Equation

This work is devoted to increasing the computational efficiency of numerical methods for the one-way Helmholtz Equation (higher-order parabolic equation) in a heterogeneous underwater environment. The finite-difference rational Padé approximation of the propagation operator is considered, whose artificial computational parameters are the grid cell sizes and reference sound speed. The relationship between the parameters of the propagation medium and the artificial computational parameters is established. An optimized method for automatic determination of the artificial computational parameters is proposed. The optimization method makes it possible to account for any propagation angle and arbitrary variations in refractive index. The numerical simulation results confirm the adequacy and efficiency of the proposed approach. Automating the selection process of the computational parameters makes it possible to eliminate human errors and avoid excessive consumption of computational resources.

Fast spatial behavior in higher order in time equations and systems

In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before.

Global existence and exponential decay of solutions for higher-order parabolic equation with logarithmic nonlinearity

Global existence and exponential decay of solutions for higher-order parabolic equation with logarithmic nonlinearity

Averaging of Higher-Order Parabolic Equations with Periodic Coefficients

In L2(Rd;Cn), we consider a wide class of matrix elliptic operators A of order 2p (where p2) with periodic rapidly oscillating coefficients (depending on x/). Here 0 is a small parameter. We study the behavior of the operator exponent e-A for 0 and small . We show that the operatore-A converges as 0 in the operator norm in L2(Rd;Cn) to the exponent e-A0 of the effective operator A0. Also we obtain an approximation of the operator exponent e-A in the norm of operators acting from L2(Rd;Cn) to the Sobolev space Hp(Rd; Cn). We derive estimates of errors of these approximations depending on two parameters: and . For a fixed 0 the errors have the exact order O(). We use the results to study the behavior of a solution of the Cauchy problem for the parabolic equation u(x,)= -(A u)(x,)+F(x,) in Rd.

BLOW UP AT INFINITY OF WEAK SOLUTIONS FOR A HIGHER-ORDER PARABOLIC EQUATION WITH LOGARITHMIC NONLINEARITY

The main goal of this work is to study the inital boundary value problem for a higher-order parabolic equation with logarithmic source term u_{t}+(-\Delta )^{m}u=uln (u). We obtain blow-up at infinity of weak solutions, by employing potential well technique. This improves and extends some previous studies.

Mass-conservative and positivity preserving second-order semi-implicit methods for high-order parabolic equations

We consider a class of finite element approximations for fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. In our approach, we first solve a variational problem and then an optimization problem to satisfy the desired physical properties of the solution such as conservation of mass, positivity (non-negativity) of solution and dissipation of energy. Furthermore, we show existence and uniqueness of the solution to the optimization problem and we prove that the methods converge to the truncation schemes [10]. We also propose new conservative truncation methods for high-order parabolic equations. A numerical convergence study is performed and a series of numerical tests are presented to show and compare the efficiency and robustness of the different schemes.

Нелокальные задачи с интегральным смещением для параболических уравнений высокого порядка

The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

Construction of the fundamental solution of a class of degenerate parabolic equations of high order

In the article, using the modified Levy method, a Green's function for a class of ultraparabolic equations of high order with an arbitrary number of parabolic degeneration groups is constructed. The modified Levy method is developed for high-order Kolmogorov equations with coefficients depending on all variables, while the frozen point, which is a parametrix, is chosen so that an exponential estimate of the fundamental solution and its derivatives is conveniently used.

Phase space path integral on torus for the fundamental solution of higher-order parabolic equations

We give a rigorous formulation of phase space path integrals on the torus $${\mathbb {T}}^{d} = ({\mathbb {R}} / 2\pi {\mathbb {Z}})^{d}$$ for fundamental solutions of higher-order parabolic equations. Especially, by using the pseudo-differential operators on the torus, we prove that the time slicing approximation of the phase space path integral converges on compact subsets of $${\mathbb {T}}^{d} \times {\mathbb {R}}^{d}$$ .

Study of 2m-th order parabolic equation in non-symmetric conical domains

This article is devoted to the study of a $N$-space dimensional linear high-order parabolic equation, subject to Cauchy-Dirichlet boundary conditions. The problem is set in a non-symmetric conical domain. The analysis is performed in the framework of weighted anisotropic Sobolev spaces by using the domain decomposition method.

Spectral inequality and resolvent estimate for the bi-Laplace operator

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called “clamped” boundary conditions, that is, homogeneous Dirichlet and Neumann conditions simultaneously.We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp (C \mu^{1/4}) where \mu is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application, we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator, in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.

Homotopy Regularization for a High-Order Parabolic Equation

In this work, we study the solvability of the Cauchy problem for a quasilinear degenerate high-order parabolic equation $$\begin{aligned} \left\{ \begin{array}{ll} u_t=(-1)^{m-1}\nabla \cdot (f^n(|u|)\nabla \Delta ^{m-1}u) &{} \hbox { in }{\mathbb {R}}^N\times {\mathbb {R}}_+, u(x,0)=u_0(x)&{} \hbox { in }{\mathbb {R}}^N, \end{array} \right. \end{aligned}$$with $$m\in {\mathbb {N}}, m>1$$ and $$n>0$$ a fixed exponent. Moreover, f is a continuous monotone increasing positive bounded function with $$f(0)=0$$ and the initial data $$u_0(x)$$ is bounded smooth and compactly supported. Thus, through a homotopy argument based on an analytic $$\varepsilon $$-regularization of the degenerate term $$f^n(|u|),$$ we are able to extract information about the solutions inherited from the polyharmonic equation when $$n=0$$.

On Some Classes of Nonlocal Boundary-Value Problems for Singular Parabolic Equations

We study the solvability of nonlocal boundary-value problems for singular parabolic equations of higher order in which the coefficient of the time derivative belongs to a space Lp of spatial variables and possesses a certain smoothness with respect to time. No constraints are imposed on the sign of this coefficient, i.e., the class of equations also contains parabolic equations with varying time direction. We obtain conditions guaranteeing the solvability of boundary-value problems in weighted Sobolev spaces and the uniqueness of the solutions.

Long-time Behaviour of Solution to Inverse Problem for Higher-order Parabolic Equation

We consider an inverse problem for the fourth-order parabolic equation. Long time behavior of the solution for the higher-order nonlinear inverse problem is established. Additional condition is given in the form of integral overdetermination.