Asymptotic behavior of unbounded solutions of linear Volterra integral equations

The asymptotic behavior as t → ∞ of solutions of ∝₀^tu(t ? s) dA(s) = f(t) is studied when f(t) satisfies a “o” estimate as $\text{t ” ∞}$, and A belongs to a weighted space and its Laplace-Stieltjes transform has finitely many zeros in its closed half-plane of convergence. Results for systems of integral equations as well as for integrodifferential systems are also given.     

Remarks on frequency domain methods for Volterra integral equations

The equation u(t) = ? ∫₀^tA(t ? τ) g(u(τ)) dτ + h(t), t ? 0 is studied on a Hilbert space H. A(t) is a family of bounded linear operators and g can be unbounded and nonlinear. Stability and asymptotic stability of solutions are studied. Frequency domain conditions are statements about the Laplace transform of A. An extension of the frequency domain method of Popov for H = R¹ is given. Here it is assumed that g is the gradient of a functional G. The frequency domain conditions are related to monotonicity and convexity conditions on A thus connecting Popov's result with work of Levin and London on equations in R¹. A second result is given in which g is not assumed to be a gradient. This extends a result of Levin in R¹. The ideas are illustrated by an example of a nonlinear partial differential functional equation.     

On a nonconvolution Volterra resolvent

Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝₀^ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝₀^tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝₀^ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝₀^tr(t, s) f(s) ds maps a weighted L¹ space continuously into another weighted L¹ space, and a weighted L^∞ space into another weighted L^∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.     

Existence and asymptotic behavior of a functional differential equation in Banach space

Existence and asymptotic behavior of solutions are given for the equation u′(t) = ?A(t)u(t) + F(t,u_t) (t ? 0) and u₀ = ? ? C([?r,0]; X)  C. The space X is a Banach space; the family $\text{{A(t) ¦ 0 ? t ? T}}$ of unbounded linear operators defined on D(A) ? X → X generates a linear evolution system and F: C → X is continuous with respect to a fractional power of A(t₀) for some t₀ ? [0, T].     

Existence and boundedness of solutions of abstract nonlinear integrodifferential equations of nonconvolution type

Existence and boundedness theorems are given for solutions of nonlinear integrodifferential equations of type $\text{d}\text{dt}\text{u(t) + Bu(t) + ∝}{}_0{}^{\mathrm{t}}\text{a(t, s) Au(s) ds ? f(t) (t > 0)}$, (1.1) u(0) = u₀, Here A and B are nonlinear, possibly multivalued, operators on a Banach space W and a Hilbert space H, where W ? H. The function f (0, ∞) → H and the kernel a(t, s): $\text{R}$ × $\text{R}$ → $\text{R}$ are known functions. The results of this paper extend the results of Crandall, Londen, and Nohel [4] for equation (1.1). They assumed the kernel to be of the type a(t, s) = a(t ? s). We relax this assumption and obtain similar results. Examples of kernels satisfying the conditions we require are given in section 4.     

Singular Volterra integral equations

Existence results are presented for the singular Volterra integral equation y(t) = h(t) + ∫₀^t k(t, s) f(s, y(s)) ds, for t ∈ [0,T]. Here f may be singular at y = 0. As a consequence new results are presented for the n^th order singular initial value problem.     

Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation $\text{u′(t) ? A(t) u(t) = ?(t), t ∈ [0, T]}$, with the initial datum u(0) = x, in the space C([0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces D_A(0)(θ, ∞) and D_A(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given.     

Representation and approximation of solutions to semilinear Volterra equations with delay

In this paper we use a theorem of Crandall and Pazy to provide the product integral representation of the nonlinear evolution operator associated with solutions to the semilinear Volterra equation: x(?)(t) = W(t, τ) ?(0) + ∝_τ^tW(t, s)F(s, x_s(?)) ds.Here the kernel W(t, s) is a linear evolution operator on a Banach space X; I is an interval of the form [?r, 0] or (?∞, 0] and F is a nonlinear mapping of R × C(I, X) into X. The abstract theory is applied to examples of partial functional differential equations.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t,s)_0?s?t?T generated by the sum −(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t,s)_0?s?t?T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂_?t∈[0,T]D(A(t)+B(t)) everywhere dense in B. We obtain a special case of our formula when B(t)=0, which, in effect, allows us to reconstruct UA(t,s)_0?s?t?T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.     

Semilinear evolution equations in Banach spaces

We study the evolution equation u′(t) = Au(t) + J(u(t)), t ? 0, where e^tA is a C₀ semi-group on a Banach space E, and J is a “singular” non-linear mapping defined on a subset of E. In Sections 1 and 2 of the paper we suppress the map J and instead consider maps K_t: E → E, t > 0, which heuristically are just e^tAJ. Under certain integrability conditions on the K_t we prove existence and uniqueness of local solutions to the integral equation u(t) = e^tAφ + ∝₀^tK_{t ? s}(u(s)) ds for all φ in E, and investigate the regularity of the solutions. Conditions which insure existence of global solutions are given. In Section 3 we recover the map J from the maps K_t, and show that the generator of the semi-flow on E induced by the integral equation has dense domain. Finally, we apply these results to a large class of examples which includes polynomial perturbations to elliptic operators on a domain in Rⁿ.     

An abstract Volterra integral equation in a reflexive Banach space

This paper discusses the existence, uniqueness, and asymptotic behavior of solutions to the equation u(t) + ∝₀^ta(t ? s) Au(s) ds = f(t), where A is a maximal monotone operator mapping the reflexive Banach space V into its dual V′.     

Positive and dead core solutions of singular BVPs for third-order differential equations

The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u^″))^′=λf(t,u,u^′,u^″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.     

Density of slowly oscillating solutions of ẋ(t) = −f(x(t − 1))

We study the scalar, nonlinear Volterra integrodifferential equation (1), x′(t) + ∫_[0,t]g(x(t ? s)) dμ(s) = f(t) (t ? 0). We let g be continuous, μ positive definite, and f integrable over (0, ∞). The standard assumption on g which yields boundedness of the solutions of (1) prevents g(x) from growing faster than an exponential as x → ∞. Here we present a weaker condition on g, which does not restrict the growth rate of g(x) as x → ∞, but which still implies that the solutions of (1) are bounded. In particular, when g is nondecreasing and either nonnegative or odd, we get bounds which are independent of g.     

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

Nonlinear Volterra equations on a Hilbert space

We consider equations of the form, u(t) = ? ∝₀^tA(t ? τ)g(u(τ)) dτ + ?(t) (I) on a Hubert space $\text{H}$. A(t) is a family of bounded, linear operators on $\text{H}$ while g is a transformation on $\text{D}$_g ?$\text{H}$ which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators e^Mt and ?e^Mtsin Mt where M is a bounded, negative self-adjoint operator on $\text{H}$ satisfy these conditions. This is shown to yield stability results for differential equations of the form, $\text{Q (}\text{d}\text{dt}\text{) u = ? P (}\text{d}\text{dt}\text{) g(u(t)) + χ(t)}$, on $\text{H}$.     

Oscillation theorems for second order nonlinear differential equations

Some oscillation criteria are established for certain second order nonlinear differential equations of the form (a(t)ψ(x(t)) x^. (t))^. + p(t) x^. (t) + q(t)f(x(t)) = 0. These criteria improve upon some of the known results by Kura, Kamenev and Philos.     

Accuracy of discrete schemes for a class of abstract evolution inequalities

For a triple of Hilbert spaces {V, H, V*}, we study a discrete and a semidiscrete scheme for an evolution inclusion of the form u′(t) + A(t)u(t) + ??(t, u(t)) ? f(t), u(0) = u ₀, t ∈ (0, T], where the pair {A(t), ?(t, ·)} consists of a family of nonlinear operators from V into V* and a family of proper convex lower semicontinuous functionals with common effective domain D(?) ? V. The discrete scheme is a combination of the Galerkin method with perturbations and the implicit Euler method. Under conditions on the data providing the existence and uniqueness of the solution of the problem in the space H ¹(0, T; V) ∩ W _∞ ¹ (0, T;H), we obtain an abstract estimate for the method error in the energy norm of first-order accuracy with respect to the time increment. By way of application, we consider a problem with an obstacle inside the domain, for which we obtain an optimal estimate of the accuracy of two implicit schemes (standard and new) on the basis of the finite element method.     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

Maximal regularity for non-autonomous Schrödinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations t_∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m.