Higher-order Parabolic Equation Research Articles

Space-Time Estimates of Mild Solutions of a Class of Higher-Order Semilinear Parabolic Equations in Lp

AbstractWe establish the well-posedness of boundary value problems for a family of nonlinear higherorder parabolic equations which comprises some models of epitaxial growth and thin film theory. In order to achieve this result, we provide a unified framework for constructing local mild solutions in C

Bistatic RCS of Electrically-Large Objects Calculations with High-Order Parabolic Equation Method

A high-order parabolic equation (PE) method is applied to calculate the Bistatic RCS of two-dimensional (2-D) Electrically-large Objects. According to the shapes of objexts, Crank-Nicolson and Pade (1,0) schemes are introduced to solve the high-order PE. The numerical results demonstrate that the method not only extends the accurate angle range but also decreases the rotation times and computational time in comparison with the traditional low-order parabolic equation method.

Infinite speed of support propagation for the Derrida–Lebowitz–Speer–Spohn equation and quantum drift–diffusion models

We show that weak solutions of the Derrida–Lebowitz–Speer–Spohn (DLSS) equation display infinite speed of support propagation. We apply our method to the case of the quantum drift–diffusion equation which augments the DLSS equation with a drift term and possibly a second-order diffusion term. The proof is accomplished using weighted entropy estimates, Hardy’s inequality and a family of singular weight functions to derive a differential inequality; the differential inequality shows exponential growth of the weighted entropy, with the growth constant blowing up very fast as the singularity of the weight becomes sharper. To the best of our knowledge, this is the first example of a nonnegativity-preserving higher-order parabolic equation displaying infinite speed of support propagation.

The Quenching Set of a MEMS Capacitor in Two-Dimensional Geometries

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro–Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.

Метод Галёркина-Петрова для одномерных параболических уравнений высокого порядка в областях с меняющейся границей

Метод Галёркина-Петрова для одномерных параболических уравнений высокого порядка в областях с меняющейся границей

Global existence and non-existence for a higher-order parabolic equation with time-fractional term

This article is concerned with the Cauchy problem for the higher-order parabolic equation with a nonlocal in time nonlinearity: {ut+(−△)mu=1Γ(1−γ)∫0t(t−s)−γ∣u∣p(s)ds,(t,x)∈R+1×RN,u(0,x)=φ(x),x∈RN, where m,p>1 and 0<γ<1. By the semigroup method and the test function method, it is proved that if p>max{1+2m(2−γ)(N−2m+2mγ)+,1γ}, then the solution with small initial datum exists globally in time. If p,N and m meet some additional conditions, we can derive decay estimate ‖u(t)‖∞≤C(1+t)−σ for some positive constant σ. On the other hand, if p≤max{1+2m(2−γ)(N−2m+2mγ)+,1γ}, where if the equality holds, we need p>N/(N−2m) with N>2m. Further if the initial datum satisfies lim¯R→∞∫|x|≤Rφ(x)dx≥0, then every nontrivial weak solution does not exist globally in time.

Stability of periodic Kuramoto–Sivashinsky waves

In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of decay. The approach extends readily to other second- and higher-order parabolic equations, for example, the Cahn Hilliard equation or more general thin film models.

Regularity of solutions of boundary-value problems for high-order parabolic equations in weighted Hölder spaces

Regularity of solutions of boundary-value problems for high-order parabolic equations in weighted Hölder spaces

Blow-Up of the Solution for some Higher Order Hyperbolic and Neutral Evolution Systems

In this paper, we give some results on the blow-up behaviors of the solution to the mixed problem for some higher-order nonlinear hyperbolic and parabolic evolution equation in finite time. By introducing the “ blow-up factor ’’, we get some new conclusions, which generalize some results [4]-[5] , [6] .

Capabilities of the high-order parabolic equation to predict propagation in boundary and shear layers.

The parabolic equation (PE) has proved its capability to deal with the long-range sound propagation in the atmosphere. It also represents an attractive alternative to the ray model to handle duct propagation in high frequencies for the noise radiated by the nacelle of aero-engines. It was recently shown that the high-order parabolic equation (HOPE), based on a Padé expansion with an order of 5, increases the aperture angle of propagation compared to the standard and the wide-angle PEs, allowing prediction close to the cutoff frequency of the duct. This paper is concerned with propagation using the HOPE in heterogeneous flows, including boundary layers above a wall and in shear layers. The thickness of the boundary layer is about dozens of centimeters; while outside it, the Mach number reaches 0.5. The boundary layer effects are investigated showing the refraction effects on a range propagation of 30 m, up to 4 kHz. On both sides of the shear layer, significant differences in the directivity patterns occur. Comparisons with the Euler solutions are considered, establishing the domain of application of the HOPE on a set of flow configurations, including those beyond its theoretical limits. [Work supported by Airbus-France.]

Unique continuation and complexity of solutions to parabolic partial differential equations with Gevrey coefficients

In this paper, we provide a quantitative estimate of unique continuation (doubling property) for higher-order parabolic partial differential equations with non-analytic Gevrey coefficients. Also, a new upper bound is given on the number of zeros for the solutions with a polynomial dependence on the coefficients.

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier–Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.

Stabilization of solutions of pseudo-differential parabolic equations in unbounded domains

We consider the first mixed problem in a cylindrical domain for a pseudo-differential parabolic equation with homogeneous Dirichlet boundary conditions and a finitely supported initial function. We find upper bounds for the -norm of a solution as in terms of a geometric characteristic introduced earlier by the author for an unbounded domain , , in the case of a higher-order parabolic equation.

On nonexistence of Baras–Goldstein type for higher-order parabolic equations with singular potentials

The celebrated result by Baras and Goldstein (1984) established that the heat equation with singular inverse square potential in a smooth bounded domain Ω ⊂ R N \Omega \subset \mathbb {R}^N , N ≥ 3 N \ge 3 , such that 0 ∈ Ω 0 \in \Omega , \[ u t = Δ u + c | x | 2 u in Ω × ( 0 , T ) , u | ∂ Ω = 0 , u_t= \Delta u + \frac c{|x|^2} u \;\; \text {in} \;\; \Omega \times (0,T), \;\; u \big |_{\partial \Omega }=0, \] in the supercritical range \[ c &gt; c H a r d y ( 1 ) = ( N − 2 2 ) 2 , c &gt; c_{\mathrm {Hardy}}(1) = \big (\frac {N-2}2\big )^2, \] does not have a solution for any nontrivial L 1 L^1 initial data u 0 ( x ) ≥ 0 u_0(x) \ge 0 in Ω \Omega or for a positive measure. Namely, it was proved that a regular approximation of a possible solution by a sequence { u n ( x , t ) } \{u_n(x,t)\} of classical solutions of uniformly parabolic equations with bounded truncated potentials given by \[ V ( x ) = c | x | 2 ↦ V n ( x ) = min { c | x | 2 , n } ( n ≥ 1 ) V(x) = \frac c{|x|^2} \mapsto V_n(x)=\min \big \{ \frac c{|x|^2}, \, n \big \} \,\,\, (n \ge 1) \] diverges, and, as n → ∞ n \to \infty , \[ u n ( x , t ) → + ∞ in Ω × ( 0 , T ) . u_n(x,t) \to +\infty \quad \mbox {in} \quad \Omega \times (0,T). \] In the present paper, we reveal the connection of this “very singular” evolution with a spectrum of some “limiting” operator. The proposed approach allows us to consider more general higher-order operators (for which Hardy’s inequalities were known since Rellich, 1954) and initial data that are not necessarily positive. In particular it is established that, under some natural hypothesis, the divergence result is valid for any 2 m 2m th-order parabolic equation with singular potential \[ u t = − ( − Δ ) m u + c | x | 2 m u i n Ω × ( 0 , T ) , w h e r e c &gt; c H ( m ) , ; m ≥ 1 , u_t = -(-\Delta )^m u + \frac c{|x|^{2m}}\, u \;\; \mathrm {in} \;\; \Omega \times (0,T), \;\; \mathrm {where} \;\; c&gt;c_{\mathrm {H}}(m), \,; m \ge 1, \] with zero Dirichlet conditions on ∂ Ω \partial \Omega and for a wide class of initial data. In particular, typically, the divergence holds for any data satisfying \[ u 0 ( x ) is continuous at x = 0 and u 0 ( 0 ) &gt; 0 . u_0(x) \;\; \text {is continuous at $x=0$ and $u_0(0)&gt;0$}. \] Similar nonexistence (i.e., divergence as ε → 0 \varepsilon \to 0 ) results are also derived for time-dependent potentials ε − 2 m q ( x ε , t ε 2 m ) \varepsilon ^{-2m}q(\frac {x}{\varepsilon }, \frac {t}{\varepsilon ^{2m}}) and nonlinear reaction terms | u | p ε 2 m + | x | 2 m \frac {|u|^p}{\varepsilon ^{2m}+|x|^{2m}} with p &gt; 1 p&gt;1 . Applications to other, linear and semilinear, Schrödinger and wave PDEs are discussed.

Asymptotically self-similar global solutions for a higher-order semilinear parabolic equation

In this paper, we study the higher-order semilinear parabolic equation {u t +(-Δ) m u=a|u| p-1 u, (t,x) ∈ℝ 1 + × ℝ N , u(0,x)=ϕ(x), x∈ℝ N , where m, p>1 and a ∈ ℝ. For p>1 +2m/N, we prove that the global existence of mild solutions for small initial data with respect to some norm. Some of those solutions are proved to be asymptotic self-similar.

Multipoint problem for higher-order parabolic equations in a parallelepiped

We study the well-posedness of a problem for a Petrovskii-parabolic equation with coefficients depending on the space coordinates and with multipoint conditions with respect to the time variable. We establish conditions for the existence and uniqueness of the classical solution of the problem. For the proof of the existence of a solution of the problem, the method of divided differences is used. We prove a metric-type theorem on lower bounds for the small denominators that appear in the construction of the solution.

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

The classic problem of regularity of boundary points for higher-order partial differential equations (PDEs) is concerned. For second-order elliptic and parabolic equations, this study was completed by Wiener’s (J. Math. Phys. Mass. Inst. Tech. 3:127–146, 1924) and Petrovskii’s (Math. Ann. 109:424–444, 1934) criteria, and was extended to more general equations including quasilinear ones. Since the 1960–1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat’ev and Maz’ya. However, the higher-order parabolic ones, with infinitely oscillatory kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon: $$u_t=-u_{xxxx}\,\,\, {\rm in}\, Q_0\,=\{|x| < R(t), \,\,-1 < t < 0\},$$ where R(t) > 0 is a smooth function on [−1, 0) and R(t) → 0+ as t → 0−. The zero Dirichlet conditions on the lateral boundary of Q 0 and bounded initial data are posed: $$u = u_x = 0\,\,\, {\rm at}\, x\,\,=\, \pm R(t), \,\, -1 \le t < 0, \quad {\rm and} \quad u(x, -1)=u_0(x).$$ The boundary point (0, 0) is then regular (in Wiener’s sense) if u(0, 0−) = 0 for any data u 0, and is irregular otherwise. The proposed asymptotic blow-up approach shows that: $$\tilde R(t) = 3^{-\frac 34} \, 2^{\frac {11}4}(-t)^{\frac 14} \left[{\rm ln} |{\rm ln}(-t)|\right]^{\frac 34}$$ belongs to the regular case, while any increase of the constant $${3^{-\frac 34} \, 2^{\frac {11}4}}$$ therein leads to the irregular one. The results are based on Hermitian spectral theory of the operator $${{\bf B}^*= -D_y^{(4)}- \frac 14 \,\, y D_y}$$ in $${L^2_{\rho^*}(\mathbb{R})}$$ , where $${\rho^*(y) = {\mathrm e}^{-a |y|^{4/3}}}$$ , $${a={\rm constant} \in (0,3 \cdot 2^{-\frac {8}3})}$$ , together with typical ideas of boundary layers and blow-up matching analysis. Extensions to 2mth-order poly-harmonic equations in $${\mathbb{R}^{N}}$$ and other PDEs are discussed, and a partial survey on regularity/irregularity issues is presented.

A higher order parabolic equation for predicting in‐duct propagation in high frequencies

Complementary methods are required to predict the in‐duct propagation in a large frequency range including the linear absorption and the flow effects. Numerical methods solving the Euler's equations are pertinent for rotational flow but limited to the low frequencies. The Boundary Element Method is applicable in a large frequency range assuming a homogeneous flow. The ray‐model is valid in high frequencies but scattering effects are difficult to implement. This paper presents the capabilities of a Higher‐Order Parabolic Equation (HOPE) to handle duct propagation in the high frequency range. In contrast with the so‐called standard PE and the wide‐angle PE, the HOPE improves the accuracy of the solution due to its wider propagation aperture angle, especially close to the cutoff frequency. Several duct propagation configurations including flow and liner are considered. Using a marching algorithm, the HOPE computes in a very short CPU time the sound propagation and represents an attractive alternative to the ...

Decay of solutions of the first mixed problem for a high-order parabolic equation with minor terms

In a cylindric domain D = (0, ∞) × Ω where Ω ⊂ ℝn+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation $$u_t + ( - 1)^k D_x^k (a(x,y)D_x^k u) + \sum\limits_{i = l}^m {\sum\limits_{\left| \alpha \right| = \left| \beta \right| = i} {( - 1)^i D_y^\alpha } } (b_{\alpha \beta } (x,y)D_y^\beta u) = 0, l \leqslant m, k,l,m \in \mathbb{N},$$ is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of the new geometrical characteristics of the domain, the upper estimate of L2-norm ∥u(t)∥ of the solution to the problem is established. In particular, in domains {(x, y) ∈ ℝn+1 | x > 0, |y1| < xa}, 0 < a < q/l, under the assumption that the upper and lower symbols of the operator L are separated from zero, this estimate takes the form Open image in new window . This estimate is determined by minor terms of the equation. The sharpness of the estimate for a wide class of unbounded domains is proved in the case k = l = m = 1.

A new family of higher order nonlinear degenerate parabolic equations

There has been much investigation of higher order nonlinear degenerate equations of the form where M is a specified function and H is the quadratic first order energy functional . The energy functional arises in many physical models, but is not universal among higher order parabolic equations. Recent investigations have motivated the study of other energy functionals, such as for p ≠ 2. We undertake such a study here, proving the existence of weak solutions for appropriate boundary conditions, nonnegativity and positivity properties of solutions. Moreover, an entropy dissipation–entropy estimate for solutions of this equation is obtained. Support properties and long time behaviour of solutions are also discussed for various cases.