# On Some Properties Of Potentials Of A Linear Parabolic Equation In A Banach Space

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### KIT

ADMISSIBILITY OF UNBOUNDED OPERATORS AND WELLPOSEDNESS OF LINEAR SYSTEMS IN BANACH SPACES BERNHARD H. HAAK AND PEER CHR. KUNSTMANN Abstract. We study linear systems, described by

### STRICHARTZ ESTIMATES FOR SCHRODINGER OPERATORS WITH A NON

the continuous spectrum [13]. In many cases it is possible to prove that the linear Schr odinger evolution decomposes into a discrete sum of bound states, plus a radi-ation term that enjoys the same dispersive properties as a free wave. Our goal is to expand the class of potentials for which this behavior is known to occur.

### fractional power type nonlinearities arXiv:2104.00132v1 [math

See [23] for the study of an inverse problem associated with a fractional parabolic operator involving time-dependent magnetic and electric potentials. In this paper, we study the semilinear fractional parabolic problem ∂tu+(−∆)s u = g Ωe ×(0,T) u = 0 Rn ×{0} (1) 1

### Personal research statements and open problems

the study of basis properties of dilated functions on a segment. The theory is well developed in the L 2 (Hilbert) space setting. During the workshop I will be interested in discussing possible extensions of this theory to the case of the Banach spaces L sfor s6= 2. Natural open problems.

### REQUIREMENTS FOR STATE EXAM MATHEMATICAL AND COMPUTATIONAL

(d) Second order linear parabolic partial diﬀerential equations Bochner spaces and their basic properties, Gelfand triple, Aubin Lions lemma. Deﬁnition of the weak solution. Initial conditions. Existence of a solution via Galerkin method, uniqueness and regularity of the solution (spatial and temporal), smoothing property, maximum principle.

### MULTIDIMENSIONAL DIFFUSION PROCESS WITH PARTIAL REFLECTION ON

Theory of Stochastic Processes Vol. 16 (32), no. 1, 2010, pp. 73 83 A. F. NOVOSYADLO MULTIDIMENSIONAL DIFFUSION PROCESS WITH PARTIAL REFLECTION ON A FIXED HYPERPLANE AND WITH

### ONE-DIMENSIONAL DIFFUSIONS IN BOUNDED DOMAINS WITH A POSSIBLE

equation (3) and associated parabolic potentials. Application of this method per-mits us to obtain the integral representation of the solution of the problem (3)-(7), which can be useful in studying additional properties of the constructed process (see [5, 6]). It is necessary to note that in the present paper we generalize the result obtained

### XI INTERNATIONAL SKOROBOHATKO MATHEMATICAL CONFERENCE

SOME PROPERTIES OF THE SPECTRUM OF THE ALGEBRA OF BO-UNDED TYPE SYMMETRIC ANALYTIC FUNCTIONS 15.20{15.40 Vira Lozynska FUNCTIONAL CALCULUS IN ALGEBRA OF !{ULTRADISTRIBUTIONS 15.40{16.00 Anna Hihliuk, Andriy Zagorodnyuk ENTIRE ANALYTIC FUNCTIONS OF UNBOUNDED TYPE ON BANACH SPACES 16.00{16.20 Liubov Kopylchuk, Viktoriia Kravtsiv

### The International Conference in Functional Analysis dedicated

tion to the inhomogeneous equation with the one-dimensional Schrodinger operator in the space of quickly decreasing functions100 Enas Mohyi Shehata Soliman and Adel-Shakob or M. Sarhan , On the e ect of the xed points of some types of a certain function. 100

### Sobolev Spaces with Weights in Domains and Boundary Value

terpolation, embedding, and other properties of the spaces are studied. As an application, a certain degenerate second-order elliptic partial diﬀerential equation is considered. 1. Introduction Let G be a domain in Rd with a non-empty boundary ∂G and ρ G(x) = dist(x,∂G). For 1 ≤ p<∞ and θ∈ R deﬁne the space L p,θ(G) as follows

### NONLINEAR STABILITY OF THE 1D BOLTZMANN EQUATION IN A

Feb 01, 2018 Boltzmann equation and the Navier-Stokes equation. The Boltzmann equation and the Navier-Stokes equations share some similarity in the whole space case, for instance, in the large-time behavior of uid-like waves. However, the Boltzmann equation is semi-linear hyperbolic; while the Navier-Stokes equations are hyperbolic-parabolic. Thus the

### Dissipative parabolic equations in locally uniform spaces

The space of bounded and uniformly continuous functions BUC(IRN), was used in [32]. More recently Bessel potentials spaces Hs p (IR N) have been used in [6]. The so called locally uniform spaces have been used in [33, 34, 12, 13] and a systematic study of linear equations in such spaces was developed in [7]. Hence

### Operator Theory of Degenerate Elliptic-Parabolic Equations

by some multiple of p, T(1F, 3C) = 0 also and thus A is Fredholm of index zero. It also follows from the proposition that since there is a natural way to make differences of continuous, bounded potentials on a fixed U into a Banach space,

### Semester III Paper VI (A) Functional Analysis - I

Normed linear spaces. Banach spaces and examples, quotient spaces of a normal linear space and its completeness, equivalent norms. (15 lectures) Unit II Bounded linear transformations, Normed linear spaces of bounded linear transformations, Halm-Banach theorem. Conjugate spaces with examples,

### 2nd Meeting IST - IME - CAMGSD

the context of the smoothing properties of the system between appropriately chosen function spaces. In particular we consider global attractors with bounded fractal dimension for a semigroup governed by an abstract semilinear parabolic equation in a Banach space. As speci c applications we present a strongly damped wave

### Existence and stability of time-periodic solutions to ON

potentials Ph. A. Martin and M. Sassoli de Bianchi-Periodic solutions of a quasilinear wave equation with variable coefficients I A Rudakov-Recent citations Parabolic evolution equations in interpolation spaces: boundedness, stability, and applications Thi Ngoc Ha Vu et al-On time-periodic solutions to the Boussinesq equations in exterior domains

### ERGODIC THEORY AND THERMODYNAMIC FORMALISM

the asymptotic statistical properties of systems evolving in time that preserve an invariant measure. Systems with chaotic behavior generally possess many invariant measures, and thermodynamic formalism borrows tools from statistical mechanics to select a distinguished measure that is physically relevant.

### Entropy and Partial Diﬀerential Equations

In Chapter IV I follow Day [D] by demonstrating for certain linear second-order elliptic and parabolic PDE that various estimates are analogues of entropy concepts (e.g. the Clausius inequality). Ias well draw connections with Harnack inequalities. In Chapter V (conserva-

### HypocoercivitybasedSensitivityAnalysisandSpectral

HypocoercivitybasedSensitivityAnalysisandSpectral ∗)

### Justiﬁcation of the lattice equation for a nonlinear elliptic

recent counterparts in the theory of nonlinear parabolic systems. These works are relevant as time-independent solutions of nonlinear parabolic systems satisfy nonlinear elliptic PDEs. In particular, the stationary solutions of a nonlinear heat equation satisfy the second-order elliptic problem (1.1).

### Some generic properties of Schr odinger operators with radial

b [0;1) is a Banach space. Using our techniques, we could also study more general potentials. For example, some singularities of a 0 at r= 0 could be allowed. For the sake of simplicity, we refrain from considering such operators. Given a nonempty open set UˆRN, let D(U) be the space of smooth (real-valued) functions on Uwith compact support.

### DIFFUSION OF FLUID IN A FISSURED MEDIUM WITH MICROSTRUCTURE

Anexistence-uniqueness theory for linear problems which exploits the strong parabolic structure of the system was given in [24]. Alternatively it is possible to eliminate Uand obtain a single functional differential equation for u in the simpler space L2(), but the structure ofthe equation then obstructs the optimal parabolic type results [18].

### NULL CONTROLLABILITY FOR A CLASS OF SEMILINEAR DEGENERATE

for semilinear parabolic equations involving an inverse-square potential were studied recently in [1, 2]. Recently, Vancostenoble [27] proved some new Carleman estimates, and conse-quently null controllability results, for the following linear degenerate/singular para-bolic equation u t (x u x) x x u= 1!h; (1.3) with suitable boundary conditions.

### E cient numerical methods for elliptic and parabolic partial

Let Hbe a Hilbert space endowed with the scalar product h:;:i, which induces the norm k:k. The space of bounded, therefore continuous, linear operators is denoted by B(H), while H stands for the dual space of H, consisting of the bounded linear functionals on H. De nition 1.1 (Spectral equivalence) Let Heb a Hilbert space.

### In the course of the proof, we show that the kernel of e

of b(x)A. This implies LP regularity of solutions to the parabolic equation d,u + P£u 0. Introduction We consider the operator if = b(x)A, a special case of a second order el liptic operator in non-divergence form with bounded measurable complex coefficients. Here b G L°°(R , C), and Reb(x) > κ for some κ > 0 and almost all χ G R

### Semigroups in Banach spaces and applications to (nonlinear

Ega~na, G., Mischler, S. Uniqueness and long time asymptotic for the Keller-Segel equation : the parabolic-elliptic case. hal-00877878 Tristani, I. Boltzmann equation for granular media with thermal force in a weakly inhomogenous setting. hal-00906770 Mischler, S., Mouhot, C. Exponential Stability of slowly decaying solutions to the Kinetic-

### Some generic properties of Schr odinger operators with radial

The genericity means that in suitable topologies the potentials having the above properties form a residual set. As we explain, (P1), (P2) are prerequisites for some applications of KAM-type results to non-linear elliptic equations. Similar properties also play a role in optimal control and other problems in linear and nonlinear partial di erential

### Non-linear Schr odinger equations with singular perturbations

spectral properties of Lare known to play an important role. Aiming at investigating the non-linear equation (0.2), another fundamental problem is to determine a class of Banach spaces which are invariant by the linear ow e itL. Indeed, in suitable perturbative regimes, one expects a local well-

### Banach and Function Spaces 2015 - 九州工業大学

Some new partial answers to a 52 year old interpolation question 1020 1100 David Yost Linear extensions and their applications 1120 1150 Ryszard Pluciennik Local Kadec-Klee properties in symmetric function spaces 1150 1220 Kichi-Suke Saito*, Naoto Komuro and Ryotaro Tanaka Which Banach space has James constant √ 2? 1400 1420 Alexei Yu

### Maximal L -regularity for parabolic boundary value problems

of parabolic equation is established in time end-point case upon the Besov space as well as the optimal trace estimates. We derive the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity.

### Time-domain boundary integral equation modeling of heat

Notation. For Banach spaces X and Y, B(X;Y) will be used to denote the space of bounded linear operators from Xto Y. We use standard function space notations: Ck(I;X) for the space of ktimes continuously di erentiable functions of a real variable in the interval Iwith values in the Banach space X, L2(O) for the space of square integrable

### IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 12

are characterized using Banach space duality theory involving special vector valued Hardy spaces. Qualitative properties of optimal feedback controllers are derived. In particular, the optimum is shown to be all-pass and performance is shown to be a monotonic decreasing function of uncertainty. A numerical solution based on duality theory, leading

### WELL-POSEDNESS AND LONG-TIME BEHAVIOR FOR A CLASS OF DOUBLY

framework. The reader is also referred to [22] for some extension to Banach spaces. In particular, the results in [24] address relation (1.1) in the situation of a linearly bounded function and a convex potential W. More recently, Segatti extended some result of [24] to classes of -convex potentials W [61].

### Bang-bang property of time optimal controls of Burgers equation

a measurable set in time for linear parabolic equations, with potentials depending on both space and time variables. The proof of the bang-bang property relies on a Kakutani xed point argument. 2010 Mathematics Subject Classi cations. 49J20, 49J30, 93B07 Keywords. Burgers equation, bang-bang property, time optimal controls, observability

### Existence of Solutions for Schrodinger Evolution Equations

If # is a Banach space C I 93£) is the space of all ^-valued (^-functions, C(I 9S£) = C°{I 9&). Assumption {A.I). For some p^l, αj^l, β>l with 0^ 1/α < 1 n/2p 9 VeL^il^ + L^Irl that is, there exist V^U^{1 T) and K 2eL°°^(/ Γ) such that V{t,x) = V x (t 9 x) + V 2 (t 9 x) a.e. (ί,x)e/ Γ x R (1.5) Under the assumption (A.I), we

### Department

with a source function, which is then found as a solution of some nonlinear operatorequation. Key words: quasi-linear parabolic equation, heat potentials, boundary value

### LIST of ABSTRACTS OF THE MEETING - unibo.it

Perturbation Method for Degenerate Inverse Problems in Banach Spaces Mohammed AL Horani 1 and Angelo Favini 2 Abstract. We are concerned with an inverse problem for a degenerate linear evolution equation. We begin with the rst-order problem where both hyperbolic and parabolic cases will be considered.

### Landau equation for Coulomb potentials near Maxwellians and

Carrapatoso, M. Landau equation for very soft and Coulomb potentials near Maxwellians, submitted Kavian, M., The Fokker-Planck equation with subcritical con nement force, submitted M., Semigroups in Banach spaces - factorization approach for spectral analysis and asymptotic estimates, in progress

### McMaster University

Digital Object Identiﬁer (DOI) 10.1007/s00220-008-0640-0 Commun. Math. Phys. 284, 803 831 (2008) Communications in Mathematical Physics Justiﬁcation of the Lattice Equation

### Existence and uniqueness of solutions of Schrödinger type

[21] J.M. Rakotoson, Linear equations with variable coefﬁcients and Banach function spaces To appear. [22] J.M.Rakotoson, Regularity of avery weak solution for parabolic equations and applications Advances in Differential Equations 16 (2011) 867 894. [23] J.M. Rakotoson, New Hardy inequalities and behaviour of linear elliptic equations