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Giuseppe Floridia

Publications and source records attributed to Giuseppe Floridia.

At least 19 recordsLinked to original sources
arXiv2026

Operator approach for time-fractional evolution equations in Banach spaces

Our first main purpose is to establish a framework for initial value problems for time-fractional evolution equation of order $\alpha \in (0,1)$ in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is the same as a generator of analytic semigroup. Based on $X$-valued Laplace transforms, we establish a solution formula yielding the well-posedness for (*). In particular, we can directly treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$. Our theory is feasibly applicable to other topics such as regularity of solutions, inverse problems and control problems.

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arXiv2025

Initial boundary value problems for time-fractional evolution equations in Banach spaces

We consider an initial value problem for time-fractional evolution equation in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is common as a generator of analytic semigroup, and in particular, we can treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$ within our framework. First we construct a solution operator $(a, F) \rrrr u$ by means of $X$-valued Laplace transform, and we establish the well-posedness of (*) in classes such as weak solution and strong solutions. We discuss also mild solutions local in time for semilinear time-fractional evolution equations. Finally we apply the result on the well-posedness to an inverse problem of determining an initial value and we establish the uniqueness for the inverse problem.

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arXiv2024

Blowup in $L^1(Ω)$-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $Ω$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blowup of solutions $u(x,t)$ in the sense that $\|u(\,\cdot\,,t)\|_{L^1(Ω)}$ tends to $\infty$ as $t$ approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of $0<p<1$, we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

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arXiv2021

Observability inequalities for degenerate transport equations

In this paper we prove an observability inequality for a degenerate transport equation. First we introduce a local in time Carleman estimate for the degenerate equation, then we apply it to obtain a global in time observability inequality by using also an energy estimate.

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arXiv2020

Inverse problems for first-order hyperbolic equations with time-dependent coefficients

We prove global Lipschitz stability for inverse source and coefficient problems for first-order linear hyperbolic equations, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point, introduced in this paper, is the choice of the length of integral curves of a vector field generated by the principal part of the hyperbolic operator to construct a weight function for the Carleman estimate. These integral curves correspond to the characteristic curves in some cases.

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arXiv2020

Backward problems in time for fractional diffusion-wave equation

In this article, for a time-fractional diffusion-wave equation $\pppa u(x,t) = -Au(x,t)$, $0<t<T$ with fractional order $α\in (1,2)$, we consider the backward problem in time: determine $u(\cdot,t)$, $0<t<T$ by $u(\cdot,T)$ and $\ppp_tu(\cdot,T)$. We proved that there exists a countably infinite set $Λ\in (0,\infty)$ with a unique accumulation point $0$ such that the backward problem is well-posed for $T \not\in Λ$.

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arXiv2020

Well-posedness for the backward problems in time for general time-fractional diffusion equation

In this article, we consider a partial differential equation with Caputo time-derivative: $\partial_t^αu + Au = F$ where $0< α< 1$ and $u$ satisfies the zero Dirichlet boundary condition. For a non-symmetric elliptic operator $-A$ of the second order and given $F$, we prove the well-posedness for the backward problem in time and our result generalizes the existing results assuming that $A$ is symmetric. The key is the perturbation argument and the completeness of the generalized eigenfunctions of the elliptic operator $A$.

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arXiv2020

Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science

Let us consider a nonlinear degenerate reaction-diffusion equation with application to climate science. After proving that the solution remains nonnegative at any time, when the initial state is nonnegative, we prove the approximate controllability between nonnegative states at any time via multiplicative controls, that is, using as control the reaction coefficient.

math.OC↗
arXiv2020

Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states

In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied in this paper: the weakly degenerate case, that is, if the reciprocal of the diffusion coefficient is summable, and the strongly degenerate case, that is, if that reciprocal isn't summable. In our main result we show that the above systems can be steered from an initial continuous state that admits a finite number of points of sign change to a target state with the same number of changes of sign in the same order. Our method uses a recent technique introduced for uniformly parabolic equations employing the shifting of the points of sign change by making use of a finite sequence of initial-value pure diffusion pro\-blems. Our interest in degenerate reaction-diffusion equations is motivated by the study of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers model) and some models in population genetics (see, e.g., the Fleming-Viot model).

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arXiv2019

Observability inequalities for transport equations through Carleman estimates

We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d $ is a bounded domain with smooth boundary $\ppp\OOO$. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of $H$ intersect $\ppp\OOO$, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.

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arXiv2019

Inverse coefficient problems for a transport equation by local Carleman estimate

We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t)) + p(x)u(x,t) = 0$ in $\OOO \times (0,T)$ where $\OOO \subset \R^n$ is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function $H(x)$ or a real-valued function $p(x)$ by initial values and data on a subboundary of $\OOO$. Our results are conditional stability of Hölder type in a subdomain $D$ provided that the outward normal component of $H(x)$ is positive on $\ppp D \cap \ppp\OOO$. The proofs are based on a Carleman estimate where the weight function depends on $H$.

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arXiv2016

Exact controllability for quasi-linear perturbations of KdV

We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero, classical Ingham inequality and HUM method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash-Moser implicit function theorems by Hörmander.

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arXiv2015

Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign

We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state $ u^*\in H_0^1 (0,1)$, with as many changes of sign in the same order as the given initial data $ u_0\in H^1_0(0,1)$, can be approximately reached in the $ L^2 (0,1)$-norm at some time $T>0$. Our method employs shifting the points of sign change by making use of a finite sequence of initial-value pure diffusion problems.

math.AP↗
arXiv2015

Well-posedness for a class of nonlinear degenerate parabolic equations

In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces.

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arXiv2015

Well-posedness of 2D and 3D swimming models in incompressible fluids governed by Navier--Stokes equations

We introduce and investigate the wellposedness of two models describing the self-propelled motion of a "small bio-mimetic swimmer" in the 2D and 3D incompressible fluids modeled by the Navier-Stokes equations. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by the rotational and elastic forces. The swimmer employs the change of its shape, inflicted by respective explicit internal forces, as the means for self-propulsion in a surrounding medium. Similar models were previously investigated in [15]-[19] where the fluid was modeled by the liner nonstationary Stokes equations. Such models are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids and air.

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arXiv2014

Approximate controllability for nonlinear degenerate parabolic problems with bilinear control

In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy-Neumann problems. First, we will obtain embedding results for weighted Sobolev spaces, that have proved decisive in reaching well-posedness for nonlinear degenerate problems. Then, we show that the above systems can be steered in $L^2$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear piecewise static controls. Moreover, we extend the above result relaxing the sign constraint on the initial date.

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