Higher-order Parabolic Equation Research Articles

Initial value problem of a higher order parabolic equation

In the present work it is studied the initial value problem for an equation in the form $$D_t^k u = \sum\limits_{j = 1}^k {ajD_t^{k - j} ( - 1)^{m + 1} \nabla ^{2m} u + A(t)u} $$ where (A(t), 0 ≤ t ≤ T) is a family of bounded linear operators defined onC(Rn), the space of all continuous functions defined onRn with the norm $$\left\| f \right\| = \mathop {\max }\limits_{x \in R_n } \left| {f(x)} \right|,f \in C(R_n )$$

Asymptotical expansions of solutions of linear parabolic equations as t → ∞

We are concerned with the asymptotical expansion as t → ∞ of the solution of the mixed problem for the high-order parabolic equation in the external domain with the first boundary conditions. This expansion follows from the asymptotical expansion of the resolvent of the corresponding elliptic operator for small values of the spectral parameter as obtained below.

Behavior of generalized solutions of mixed problems for quasilinear parabolic equations of high order in unbounded domains

Behavior of generalized solutions of mixed problems for quasilinear parabolic equations of high order in unbounded domains

The Cauchy problem for time-dependent abstract parabolic equations of higher order

After the pioneering works by Chazarain [2] and Dubinskii [3] in the early seventies, several papers were devoted to higher order differential equations in a Banach space, especially for either constant coefficient or second order equations. Detailed references can be find in [4, Sect. 25(c)]. Very recent results in the constant coefficient case are due to Sandefur [S] for second order semilinear equations and to Neubrander [S] for linear arbitrary order equations. In a recent paper [7], we considered an equation such as

A stochastic solution of a high order parabolic equation

On considere le probleme aux valeurs initiales ∂W(t, x)/∂t=(A+B)W(t, x), t>0, x∈R d , W(o, x)=f(x) que l'on resout par une methode stochastique

A splitting procedure for parabolic equations of higher order

In this paper, the numerical solution of parabolic equations of order n is shown to be easily accomplished by a splitting procedure involving the use of n computational nets.

Construction of a fundamental solution for a class of high-order degenerate parabolic equations

Construction of a fundamental solution for a class of high-order degenerate parabolic equations

Central Limit Theorem for Signed Distributions

This paper contains an improved version of existing generalized central limit theorems for convergence of normalized sums of independent random variables distributed by a signed measure. It is shown that under reasonable conditions, the normalized sums converge in distribution to “higher-order” analogues of the standard normal random variable, in the sense that the density of the limiting signed distribution is the fundamental solution of a higher-order parabolic partial differential equation that is a generalization of the heat equation.

Central limit theorem for signed distributions

This paper contains an improved version of existing generalized central limit theorems for convergence of normalized sums of independent random variables distributed by a signed measure. It is shown that under reasonable conditions, the normalized sums converge in distribution to “higher-order” analogues of the standard normal random variable, in the sense that the density of the limiting signed distribution is the fundamental solution of a higher-order parabolic partial differential equation that is a generalization of the heat equation.

Stochastic parabolic equations of higher order in t

Stochastic parabolic equations of higher order in t

On the solution of boundary value problems for linear parabolic equations of higher order

On the solution of boundary value problems for linear parabolic equations of higher order

On equations of evolution and parabolic equations of higher order in t

On equations of evolution and parabolic equations of higher order in t

A Perturbation Series for Cauchy's Problem for Higher-Order Abstract Parabolic Equations

The application of Phillips' perturbation theorem to Cauchy's problem for higher-order parabolic equations is justified by an argument from the theory of Fourier transforms of entire functions.

Finite-Resistivity Stabilities of a Sheet Pinch

A simple but general approach to obtain the stability criteria is presented through an elementary proof of the uniqueness theorem for the higher order parabolic equations by constructing an energy inequality. The theorem is applied to the isothermal, resistive, viscous and incompressible magnetohy-drodynamical (MHD) equations of a sheet pinch where we reduce the equations into an equation governing the two dimensional vorticity and an equation governing the parallel motion with respect to the acting force. The theorem yields the conditions for the stable parallel motion and for the conservation of vorticity. The result is compared with other works and the effects of viscosity and resistivity are discussed.

A Property of Weak Solutions for Some Parabolic Equations of Higher Order

Recently Aronson [1] proved the uniqueness property of weak solutions of the initial boundary value problem for second order parabolic equations with discontinuous coefficients. An analogous result to Aronson’s was proved by Kuroda M in the case of some parabolic equations of higher order, where the method due to Aronson [2] plays an essential role. In this paper, under the same idea we shall be concerned with the asymptotic behavior of weak solutions for parabolic equations of higher order of the divergence form, when the data are prescribed on a portion of a time-like surface.

Asymptotic behavior of solutions of parabolic equations of higher order

Asymptotic behavior of solutions of parabolic equations of higher order

Unique Continuation for Parabolic Equations of Higher Order

Let x = (xl,…xn) be a point in the n-dimensional Euclidean space and let be the unit sphere In the (n + 1)-dimensional Euclidean space with coordinate (x, t), we putandwhere denotes the boundary of . We also use the following notation:

Unique Continuation Property of Non-Positive Weak Subsolutions for Parabolic Equations of Higher Order

When L is a parabolic differential operator of second order, Nirenberg [6] proved the maximum principle for the function u which has second order continuous derivatives and satisfies Lu≧0. Recently Friedman [2] has proved the maximum principle for the measurable function satisfying Lu≧O in the wide sense. This function is named a weakly L-subparabolic function. On the other hand, Littman [5] earlier than Friedman, has defined a weakly A- subharmonic function for an elliptic differential operator A of second order and has showed the maximum principle for it.

On the Dirichlet problem for certain higher order parabolic equations

On the Dirichlet problem for certain higher order parabolic equations