Solution Semigroup Research Articles

Differentiability of the age-dependent population system with time delay in the birth process

In this paper, by using semigroup approach, we concerned with the regularity of the age-dependent population system with instantaneous time delay in the birth process. We prove that, under some additional condition, the solution semigroup { T ( t ) , t ⩾ 0 } of the population evolution equation is eventually differentiable, i.e., there exists t 0 > 0 such that T ( t ) is differentiable for t > t 0 .

Holomorphic solution semigroups for Sobolev-type equations in locally convex spaces

The problem of the existence of an exponentially bounded solution semigroup strongly holomorphic in a sector is studied for a Sobolev-type linear equation (1) with continuous operator , , and closed densely defined operator , where and are sequentially complete locally convex spaces. It is shown that the condition of the -sectoriality of the operator , which generalizes the well known condition of sectoriality, is necessary and sufficient for the existence of such semigroups degenerate at the -associated vectors of the operator of height and lower and the existence of pairs of invariant subspaces of the operators and . Generalizations of Yosida's theorem and results on the existence of a holomorphic solution semigroup for equation (1) in Banach spaces are obtained. These results are used in the study of the weakened Cauchy problem for equation (1) and for the corresponding non-linear equation. One application of the abstract results is a theorem on sufficient conditions for the solubility of the Cauchy problem for a class of equations in Fréchet spaces of a special kind. It is used in the analysis of the periodic Cauchy problem for a partial differential equation with displacement not solved with respect to the time derivative.

Asymptotic Behaviour of Parabolic Problems with Delays in the Highest Order Derivatives

We use semigroup methods to investigate the partial functional differential equation $u’(t)=Au(t)+ \int_{-r}^0 dB(\theta)u(t+\theta)$ for a sectorial operator $A$ on a Banach space $X$ and a function $B:[-r,0] \to\cL(D(A),X)$ of bounded variation having no mass at 0. Using a perturbation theorem due to Weiss and Staffans, we construct the solution semigroup on a product space in order to solve the delay equation in a classical sense. Employing the spectrum of the semigroup and its generator, we then study exponential dichotomy and stability of solutions. If $X$ is a Hilbert space, %C% these properties can be characterized by estimates on $(\la-A-\widehat{dB}(\la))^{-1}\in\cL(X,D(A))$. Related results on stability also hold for general Banach spaces. The case $B=\eta A$ with scalar valued $\eta$ is treated in some detail.

CONSERVATIVE AND SHATTERING SOLUTIONS FOR SOME CLASSES OF FRAGMENTATION MODELS

Equations describing processes of cluster fragmentation have received considerable attention in recent years due to their importance in modelling e.g. the polymer degradation, droplet break-up, or rock crushing. In this paper we shall consider two mathematically most interesting features of them. One of these is the so-called shattering fragmentation, that is, the decrease of the total mass of the system that is formally conservative. The other is the existence of multiple solutions which indicates that in some cases the equations do not give the full picture of the dynamics of the model. Shattering fragmentation, that is considered to be an analogous but opposite process to a better understood gelation in coagulation models, has been investigated in several papers whose interest was, however, restricted to coefficients of a particular form that allowed either explicit solutions or yielded to a probabilistic approach. There have been few attempts to provide a deeper analysis of the existence of multiple solutions; most authors confined themselves to noticing them. In the present paper we present a systematic approach to both problems using the semigroup theory, and link them to the characterization of the generator of the solution semigroup of the fragmentation equation. As a result we provide criteria for both existence and absence of shattering fragmentation as well as of multiple solutions for large classes of coefficients that include those considered in all earlier works.

Well-posedness of linear partial differential equations with unbounded delay operators

In this paper we show the well-posedness of the following constant delay equation: u ̇ (t)=Au(t)+Bu(t−r), t>0, u(0)=x, u(θ)=f(θ), θ∈[−r,0], where ( A, D( A)) generates a strongly continuous semigroup ( T( t)) t⩾0 on a Banach space X and ( B, D( B)) is an unbounded and closed linear operator on X. Our approach is to transform the delay equation into an abstract Cauchy problem in a special phase space and apply semigroup theory. Furthermore, we show that the solution semigroup corresponding to the above delay equation is uniformly exponentially bounded. Finally, we give an example to explain the well-posedness of delay equations.

Hyperbolic Equations in Uniform Spaces

The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of ${\mathbb R}^n$. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space $\dot H^1_{\rm lu}(

Asymptotic smoothing effect of solutions to weakly dissipative Klein–Gordon–Schrödinger equations

In this paper the authors consider the Cauchy problem of weakly dissipative Klein–Gordon–Schrödinger equations through Yukawa coupling in R 3 . Making use of a Strichartz type inequality and a suitable decomposition of the solution semigroup they prove the asymptotic smoothing effect of the solutions.

Positivity and stability for some partial functional differential equations

The aim of this work is to study the stability for some linear partial functional differential equations. We assume that the linear part is non-densely defined and satisfies the Hille-Yosida condition. Using the positiveness, we give nessecary and sufficient conditions independently of the delay to ensure the uniform exponential stability of the solution semigroup. An application is given for a reaction diffusion equation with several delays.

Norm continuity and stability for a functional differential equation in Hilbert space

A functional differential equation in Hilbert space with initial data on [−h,0] is considered. An unbounded operator A and a square integrable weight function are acting in the distributed delay term. For a not necessarily continuous weight function the norm continuity of the associated solution semigroup is established at every t>h. In the case that the spectrum of A is real and negative, the asymptotic stability of the solution is obtained.

Stability conditions for abstract functional differential equations in Hilbert space

Stability conditions for functional differential equations of the form: du(t)/dt = Au(t) + bAu(t − h) + (a* Au)(t) are studied, where A is the infinitesimal generator of an analytic semigroup in a Hilbert space, b ≠ 0 and the convolution term contains a square integrable real function a ≠ 0. Norm discontinuity of the solution semigroup of the equation with discrete delay is avoided by studying the inverse of the characteristic operator. Sufficient and necessary conditions for the uniform exponential stability of the solution semigroup are obtained. The results are applied to a retarded partial integrodifferential equation.

PFDE with nonautonomous past

We study Cauchy problems associated topartial differential equations with infinite delay where thehistory function is modified by an evolution family. Usingsophisticated tools from semigroup theory such as evolutionsemigroups, extrapolation spaces, or the critical spectrum, weprove well-posedness and characterize the asymptotic behavior ofthe solution semigroup by an operator-valued characteristicequation.

Nonlinear homogeneous retarded differential equations and population dynamics via translation semigroups

In this paper nonlinear homogeneous retarded differential equations and models from population dynamics and epidemiology are considered and studied by the use of translation semigroup theory. We show that the corresponding solution semigroups are equivalent to appropriate translation semigroups. Existence results for the retarded equations are established by taking the space of initial functions of the form W 1,p ((-r,0),F) where F is a Banach space, 1≤ p<∈fty and 0<r\le∈fty .

Uniqueness of Classical and Nonclassical Solutions for Nonlinear Hyperbolic Systems

In this paper we establish a general uniqueness theorem for nonlinear hyperbolic systems of partial differential equations in one-space dimension. First of all we introduce a new notion of admissible solutions based on prescribed sets of admissible discontinuitiesΦ and admissible speedsψ. Our definition unifies in a single framework the various notions of entropy solutions known for hyperbolic systems of conservation laws, as well as for systems in nonconservative form. For instance, it covers the nonclassical (undercompressive) shock waves generated by a vanishing diffusion-dispersion regularization and characterized via a kinetic relation. It also covers Dal Maso, LeFloch, and Murat's definition of weak solutions of nonconservative systems. Under certain natural assumptions on the prescribed sets Φ and ψ and assuming the existence of a L1-continuous semi-group of admissible solutions, we prove that, for each Cauchy datum at t=0, there exists at most one admissible solution to the Cauchy problem depending L1-continuously upon the initial data. In particular, our result shows the uniqueness of the L1-continuous semi-group of admissible solutions. In short, this paper proves that supplementing a hyperbolic system with the “dynamics” of elementary discontinuities characterizes at most one L1-continuous and admissible solution.

Stability of $L^\infty$ Solutions for Hyperbolic Systems with Coinciding Shocks and Rarefactions

We consider a hyperbolic system of conservation laws \[ \left\{ \begin{array}{c} u_t + f(u)_x = 0, u(0,\cdot) = u_0, \end{array} \right. \] where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates w, we prove that there exists a semigroup of solutions $u(t) = \mathcal{S}_t u_0$, defined on initial data $u_0 \in L^\infty$. The semigroup $\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{{\rm loc}}$ topology. Moreover, $\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.

A renorming method for thermoelastic models

We consider the coupled partial differential equations which arise in modeling linear thermoelastic structures. When this model is reformulated as an abstract Cauchy problem, it is known that the resulting solution semigroup is exponentially stable, and in certain cases it is analytic. In most cases, though, the infinitesimal generator of the semigroup does not satisfy a strict dissipative inequality. We construct a new norm, equivalent to the energy norm, for which a strict dissipative inequality is obtained. In the case of an analytic semigroup, the new norm allows the semigroup generator to become associated with a coercive esquilinear form.

Gevrey Regularity for the Attractor of a Partially Dissipative Model of Bénard Convection in a Porous Medium

Convective flow though a porous medium can be modeled by Darcy's law—a linear, weakly damped momentum equation—coupled with an advection–diffusion equation for the energy. The solution semigroup for this system is not smoothing, and the solution of the momentum equation does not gain regularity with respect to its initial value in finite time. However, it is known that the semigroup is asymptotically smoothing, so that the system possesses a finite dimensional global attractor as well as exponential attractors. We show that the global attractor is contained in a special Gevrey class of regularity and, in particular, is real analytic. The key idea is the use of a Fourier splitting method to approximate every orbit asymptotically in time by a Gevrey-regular function.

A uniqueness condition for hyperbolic systems of conservation laws

Consider the Cauchy problem for a hyperbolic $n\times n$ system of conservation laws in one space dimension: <p align="center"> $u_t + f(u)_x = 0$, $u(0,x)=\bar u (x).$ $(CP)$ <p align="left" class="times"> Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of $(CP)$ is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves.

Stability of $L^\infty$ solutions of Temple class systems

Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of ${{\bf L}}^\infty$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the ${{\bf L}}^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

L^1 stability of patterns of non-interacting large shock waves

We consider a strictly hyperbolic n × n system of conservation laws in one space dimension ut + f(u)x = 0, together with Cauchy initial data u(0, x) = ū(x), that is a small BV ∩ L perturbation of fixed Riemann data (u− 0 , u 0 ). We a priori assume that the latter problem is solved by M large shocks (2 ≤ M ≤ n) of different characteristic families, each of them Majda stable and Lax compressive. We prove that under a suitable Finiteness Condition the problem has a unique solution defined globally in space and time, while a stronger Stability Condition guarantees the existence of a Lipschitz semigroup of solutions.

Oleinik Type Estimates and Uniqueness for n×n Conservation Laws

Let ut+f(u)x=0 be a strictly hyperbolic n×n system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of u in a forward neighborhood of each point in the t-x plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleinik in the scalar case.