Abstract Parabolic Equations Research Articles

Solvability on the real line of a class of linear volterra integrodifferential equations of parabolic type

We consider an abstract parabolic integrodifferential equation with infinite delay in general Banach space X: $$u'(t) = Au(t) + \int\limits_{ - \infty }^t {K(t - s)u(s)ds + f(t),t \in R}$$ (*) where A: D(A) ⊂ X → X generates an analytic semigroup, and K(t) e L(D(A), X) for every t ⩾ 0. Under suitable assumptions on the kernel K, we extend to equation (*) the well known results about bounded solutions, periodic solutions, and solutions with exponential growth, of the abstract parabolic equation: $$v'(t) = Av(t) + f(t),t \in R.$$ (**)

Book Review: The equations of Navier-Stokes and abstract parabolic equations

Book Review: The equations of Navier-Stokes and abstract parabolic equations

On the existence of periodic solutions to nonlinear abstract parabolic equations

On considere l'equation parabolique non lineaire dans un espace de Hilbert reel H, qui est de la forme du(t)/dt+∂φ(u(t))→f(t), ou f∈L loc 2 (R;H), φ est une fonctionnelle convexe semi-continue inferieurement sur H et ∂φ est la sous-differentielle de φ. On montre l'existence des solutions antiperiodiques sous une condition differente de la coercivite

Bounded solutions of linear periodic abstract parabolic equations

SynopsisWe study the asymptotic behaviour of the parabolic evolution operatorG(t,s) associated with a family of generators of analytic semigroups {A(t);t∊ ℝ} in general Banach space, under the periodicity assumptionA(t) =A(t+T). We find new maximal regularity results for the bounded solutions of abstract nonhomogeneous parabolic equations in unbounded time intervals.

On the evolution operator for abstract parabolic equations

We find a new construction of the evolution operatorG(t, s) associated to a family {A(t), 0≦t≦T} of generators of analytic semigroups in a Banach spaceX. We study the dependence ofG (t, s) ont ands, and we give regularity results for the solution of the i.v.p.u′(t)=A(t)u(t)+f(t),u(0)=x.

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

Time analyticity for solutions of nonlinear abstract parabolic evolution equations

We study analyticity with respect to time of the solutions to a class of nonlinear evolution equations in Banach space. The results are applied to Cauchy-Dirichlet problems for fully nonlinear parabolic integrodifferential equations such as u t(t, x) = f(t, x, u(t, x), ▽u(t, x), Δu(t, x)) + ∝ 0 t g(t, s, x, u(s, x), ▽u(s, x), Δu(s, c))ds, t ⩾ 0, x ϵ \\ ̄ gW , where Ω is a regular bounded open set in R n .

Evolution operators for higher order abstract parabolic equations

Evolution operators for higher order abstract parabolic equations

A remark on semilinear perturbations of abstract parabolic equations

Abstract : Abstract results are given which apply to semilinear parabolic equations. Simple examples are provided by initial-value problems for semilinear equation like u sub t = delta u + f(u). As applied to this situation, the results show that for suitable f and t greater than 0 the solution u(t,phi) of the problem with the initial-value phi varies continuously together with its space derivatives as phi varied in a weaker sense - e.g. small perturbations of phi in some L sub P-norms cause small changes in u(t,phi) in the space on continuously differentiable functions. (Author)

Bionomics of hill-stream cyprinids. IV. Length-weight relationship ofLabeo dero (Ham.) from India

Parabolic equations describing the body length-body weight relationships in 311Labeo dero (Ham.) are determined. The differences between regression coefficients of the two sexes and length classes were highly significant. The difference between regression coefficients of the two length classes was significant at 5% level.

Analyticity of the maximal solution of an abstract nonlinear parabolic equation

where u(t, x): [0, 7’j X R” + R satisfies other suitable conditions. Problem (P) is studied under the assumption that af/au generates an analytic semigroup in E and has domain F, which corresponds to a parabolicity condition on equation (P’). Problems of this kind have been studied by monotonicity methods (see Brezis 111, Crandall & Ligget [3] and Crandall & Pazy [4]) to obtain global but weak solutions of (P). Here we find a maximally defined strict solution by means of a linearization method and by virtue of maximal regularity results for the linear case. This method has been used by Da Prato & Grisvard in [5] to obtain strict solutions of(P); with arguments which seem to be more simple and suitable to our purpose we get also the analyticity of u with respect to (1, W) when f is analytic. In section 1 some notation and assumptions are given. Problem (P) is studied in section 2 where we prove existence and uniqueness of the strict solution of (P), which is defined on a maximal time interval contained in [0, T] and it is continuous with respect to (t, ~0). In section 3 we give conditions to get global existence of the solution. In section 4 we study analyticity with respect to (t, ug) of the maximally defined solution of (P) and in section 5 a simple example is given. In a subsequent paper we will show how this theory applies to the study of the analyttctty of the solutions of nonlinear parabol.ic partial differential equations in spaces of Holder continuous functions.

Hölder regularity for non-autonomous abstract parabolic equations

This paper contains some existence and uniqueness results for the strict and classical solutionsu : [0,T] →E of the non-autonomous evolution equationu 1(t)=Λ(t)u(t)+f(t) in a Banach spaceE under the classical Tanabe-Sobolevski assumptions. These results do not require use of the fundamental solution and give new information about the holder-regularity of the solutions.

Stabilization of solutions of abstract parabolic equations

Stabilization of solutions of abstract parabolic equations

Asymptotic equivalence of abstract parabolic equations with delays

Asymptotic equivalence of abstract parabolic equations with delays

The averaging method and bifurcation of bounded solutions of abstract parabolic equations

The averaging method and bifurcation of bounded solutions of abstract parabolic equations

Two point problems and analyticity of solutions of abstract parabolic equations

Fort ∈ [a, b], letA(t) be the unbounded operator inH 0,p (G) associated with an elliptic-boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ ≥0. Existence and uniqueness results are obtained for weak and strict solutions of two-point problems of the type (du/dt)−A(t) u(t) =f(t),E 1(α)u (α)=u α,E 2 (β)u (β)=u β. Here [α, β) χ- [a, b],E 1 (α) andE 2 (β) are spectral projections associated withA(α) andA(β) respectively, andA(α)E 1 (α) and =A (β)E 2 (β) are infinitesimal generators of analytic semigroups. WhenA(t) andf(t) are analytic in a convex, complex neighborhoodO of [a, b] we show that for someθ i ,i=1,2, any solution ofdu/dt =A(t)u (t)=f(t) in [a, b] is analytic and satisfies the above equation in the setO∩{t; t ≠ a, t ≠ b, | arg (t −a) | <θ 1, | arg (b −t) |θ 2}.

Regularity properties of solutions of some abstract parabolic equations

Consider the equation (i) (da/dt)—A(t)u(t)=f(t) where fort ∈ [a, b],A(t) is a densely defined and closed linear operator in a Banach spaceX. Assume the existence of bounded projectionsE i(t),i=1, 2, such thatA(t) E 1(t) and —A(t)E 2(t) are infinitesimal generators of analytic semigroups andA(t) is completely reduced by the direct sum decompositionX = Σ i b = 1/2 ⊕E i (t)X. We show that any solutionu(t) of (i) is inC ∞(a, b) and satisfies the inequalities (1.2) provided thatf(t) andA(t) are infinitely differentiable in [a, b] in a suitable sense. In caseA(t) andf(t) are in a Gevrey class determined by the constants {M n} we have (1. 3). Applications are given to the study of solution of (i) where fort ∈ [a, b]A(t) is the unbounded operator inH 0,p (G) associated with an elliptic boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ > 0.

HIGHER APPROXIMATIONS OF THE AVERAGING METHOD FOR ABSTRACT PARABOLIC EQUATIONS

In the authors's article in this journal, volume 10 (1970), pp. 51-59, it was shown that the averaging method of N. N. Bogoljubov can be applied to abstract parabolic equations with unbounded operators which generate analytic semigroups. The nearness of the solutions of the original and the averaged equations in a certain norm was proved.In the present article it is shown that the successive approximations of the averaging method are near the solution of the original equation in stronger norms and in a stronger asymptotic sense.Bibliography. 7 items.

New Integral Representations for Solutions of Cauchy's Problem for Abstract Parabolic Equations

A study has been made of Cauchy's problem for a class of abstract parabolic differential equations. Our study is based upon a transformation that maps solutions of second-order abstract Cauchy problems into solutions of first-order. This is a preliminary report on some of the results obtained. These results include (1) new representations for solutions of abstract parabolic equations, (2) applications to the heat equation, (3) a discussion of solution representations for second-order abstract Cauchy problems, and (4) applications to nonwell posed (in the sense of Hadamard) Cauchy problems.

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.