DCDS
On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces
Simona Fornaro Abdelaziz Rhandi
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5049-5058 doi: 10.3934/dcds.2013.33.5049
In this paper we give sufficient conditions ensuring that the space of test functions $C_c^{\infty}(R^N)$ is a core for the operator $$L_0u=\Delta u-Mx\cdot \nabla u+\frac{\alpha}{|x|^2}u=:Lu+\frac{\alpha}{|x|^2}u,$$ and $L_0$ with domain $W_\mu^{2,p}(R^N)$ generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p_\mu(R^N),\,1 < p < \infty$. Here $M$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$. The proofs are based on an $L^p$-weighted Hardy's inequality and perturbation techniques.
keywords: Ornstein-Uhlenbeck operator. core dissipative and dispersive operator Hardy's inequality Inverse square potential positivity preserving $C_0$-semigroup
DCDS
Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$
Abdelaziz Rhandi Roland Schnaubelt
Discrete & Continuous Dynamical Systems - A 1999, 5(3): 663-683 doi: 10.3934/dcds.1999.5.663
We prove existence and uniquences of positive solutions of an age-structured population equation of McKendrick type with spatial diffusion in $L^1$. The coefficients may depend on age and position. Moreover, the mortality rate is allowed to be unbounded and the fertility rate is time dependent. In the time periodic case, we estimate the essential spectral radius of the monodromy operator which gives information on the asymptotic behaviour of solutions. Our work extends previous results in [19], [24], [30], and [31] to the non-autonomous situation. We use the theory of evolution semigroups and extrapolation spaces.
keywords: resolvent positive operator Hillc-Yosida operator Age-structured population equation irreducibility quasi-compact semigroup essential spectral radius balanced exponential growth Miyadera perturbaion extrapolated semigroup positive evolution family evolution semigroup.
DCDS-S
On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials
Tiziana Durante Abdelaziz Rhandi
Discrete & Continuous Dynamical Systems - S 2013, 6(3): 649-655 doi: 10.3934/dcdss.2013.6.649
In this note we give sufficient conditions for the essential self-adjointness of some Kolmogorov operators perturbed by singular potentials. As an application we show that the space of test functions $C_c^∞(R^N \backslash \{0\})$ is a core for the operator $Au= Δu-Bx∇u+\frac{c}{|x|^2} u=:Lu+\frac{c}{|x|^2} u, u ∈ C_c^∞(R^N \backslash \{0\}),$ in $L^2(R^N,\mu)$ provided that $c\le \frac{(N-2)^2}{4}-1$. Here $B$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$.
keywords: positivity preserving $C_0$-semigroup Ornstein-Uhlenbeck operator Inverse square potential Hardy's inequality essential self-adjointness Kolmogorov operator.
CPAA
Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces
Simona Fornaro Federica Gregorio Abdelaziz Rhandi
Communications on Pure & Applied Analysis 2016, 15(6): 2357-2372 doi: 10.3934/cpaa.2016040
In this paper we give sufficient conditions on $\alpha \ge 0$ and $c\in R$ ensuring that the space of test functions $C_c^\infty(R^N)$ is a core for the operator \begin{eqnarray} L_0u=(1+|x|^\alpha )\Delta u+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u, \end{eqnarray} and $L_0$ with a suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(R^N), 1 < p < \infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.
keywords: positivity preserving $C_0$-semigroup core unbounded diffusion. Inverse square potential Hardy's inequality dissipative and dispersive operator
DCDS
Unbounded perturbations of the generator domain
Said Hadd Rosanna Manzo Abdelaziz Rhandi
Discrete & Continuous Dynamical Systems - A 2015, 35(2): 703-723 doi: 10.3934/dcds.2015.35.703
Let $X,U$ and $Z$ be Banach spaces such that $Z\subset X$ (with continuous and dense embedding), $L:Z\to X$ be a closed linear operator and consider closed linear operators $G,M:Z\to U$. Putting conditions on $G$ and $M$ we show that the operator $\mathcal{A}=L$ with domain $D(\mathcal{A})=\big\{z\in Z:Gz=Mz\big\}$ generates a $C_0$-semigroup on $X$. Moreover, we give a variation of constants formula for the solution of the following inhomogeneous problem \begin{align*} \begin{cases} \dot{z}(t)=L z(t)+f(t),& t\ge 0,\cr G z(t)=Mz(t)+g(t),& t\ge 0,\cr z(0)=z^0. \end{cases} \end{align*} Several examples will be given, in particular a heat equation with distributed unbounded delay at the boundary condition.
keywords: unbounded perturbation closed-loop systems $C_0$--semigroup inhomogeneous boundary problem. regular linear systems Banach space
DCDS-S
Kolmogorov equations perturbed by an inverse-square potential
Gisèle Ruiz Goldstein Jerome A. Goldstein Abdelaziz Rhandi
Discrete & Continuous Dynamical Systems - S 2011, 4(3): 623-630 doi: 10.3934/dcdss.2011.4.623
In this paper we present a nonexistence result of exponentially bounded positive solutions to a parabolic equation of Kolmogorov type with a more general drift term perturbed by an inverse square potential. This result generalizes the one obtained in [8]. Next we introduce some classes of nonlinear operators, related to the filtration operators and the $p$-Laplacian, and involving Kolmogorov operators. We establish the maximal monotonicity of some of these operators. In the third part we discuss the possibility of some nonexistence results in the context of singular potential perturbations of these nonlinear operators.
keywords: nonlinear parabolic equations Hardy's inequality positive solutions $p$-Kolmogorov operator. Inverse square potential Ornstein-Uhlenbeck operator critical constant

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