Giuseppe Floridia
In this paper we consider a multidimensional semilinear reaction-diffusion equation and we obtain at any arbitrary time an approximate controllability result between nonnegative states using as control term the reaction coefficient, that is via multiplicative controls.
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Stackelberg-Nash strategy for the null controllability of semilinear degenerate equations in non-cylindrical domains
本研究针对非柱形域内一维半线性退化抛物方程的零能控问题，采用Stackelberg-Nash策略，结合退化非自治系统的Carleman不等式与Liusternik逆函数定理，证明了该系统的局部零能控性。
Local controllability of free boundary three-dimensional semilinear radial parabolic equations
针对具有Stefan边界条件和径向对称数据的自由边界三维半线性热方程，通过简化为一维形式并适配Carleman不等式，结合Schauder不动点定理证明其局部零可控性，为该问题在一维以上空间维度提供首个可控性结果。
Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms
本文研究了具有非局部项的半线性抛物型方程组的可控性问题，通过结合不动点定理和线性化动力学估计，建立了在固定时间内的局部零可控性，并拓展了线性系统的可控性范围。
The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $Δ^{\fracα{2}} u=ρ(x)\,φ(u)$ admits nonnegative nontrivial bounded solutions in the whole space $\mathbb{R}^N$.
We consider the second order semilinear elliptic system $Δu= p\left( x\right) v^α,$ $Δv= q\left(x\right) u^β,$ where $x \in \mathbf{R}^N,$ $N \geq 3,$ $α$ and $β$ are positive constants, $p$ and $q$ are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions $p$ and $q$ are of slow decay.
We consider a linear nonlocal heat equation in a bounded domain $Ω\subset\mathbb{R}^d$ with Dirichlet boundary conditions. The non-locality is given by the presence of an integral kernel. We analyze the problem of controllability when the control acts on an open subset of the domain. It is by now known that the system is null-controllable when the kernel is time-independent and analytic or, in the one-dimensional case, in separated variables. In this paper, we relax this assumption and we extend the result to a more general class of kernels. Moreover, we get explicit estimates on the cost of n…
In this paper, we prove the small-time global null-controllability of forward (resp. backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and diffusion terms (resp. in the drift term). In particular, we solve the open question posed by S. Tang and X. Zhang, in 2009. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameter…
Controllability of some semilinear shadow reaction-diffusion systems
本研究针对半线性反应扩散系统，通过定义合适非线性项与改进一致Carleman估计，证明原始模型的零可控性可被阴影极限模型继承，拓展了线性情形的相关结果。
Initial traces of solutions to a semilinear heat equation under the Dirichlet boundary condition
该研究探讨Dirichlet边界条件下半线性热方程非负解初始迹的定性性质，针对Cauchy–Dirichlet问题得到非负解存在的严格充要条件，确定可解非负初值的最优奇性。
In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability ( finite-approximate controllability) for the semilinear evolution equations in Hilbert spaces. We state a useful characterization of the finite-approximate controllability for linear evolution equation in terms of resolvent-like operators. We also find a control so that, in addition to the approximate controllability requirement, it ensures finite dimensional exact controllability. Assuming the approximate controllability of the corresponding linearized equation we…
Nash-Stackelberg controllability for coupled systems of degenerate equations in non-cylindrical domains
本文研究了非圆柱域中退化方程耦合系统的纳什-斯塔克尔伯格可控性，通过分层零可控性方法和Carleman估计实现局部可控性。
In this paper, we consider a class of fractional integro-differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the approximate controllability for a class of fractional semilinear integro-differential inclusions in Hilbert spaces. We use Bohnenblust-Karlin's fixed point theorem to prove our main results. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. An example is also given …
In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, …
Let $L$ be a second order uniformly elliptic differential operator in a domain $D$ of $\mathbb{R}^{d}$, $ψ:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing continuous function and let $ξ,g:D\to\mathbb{R}_+$ be locally bounded Borel measurable functions. Under appropriate conditions, we determine a function $φ$ with values in $]0,1]$ such that for every nonnegative solution to inequality $-Lu+ξψ(u) \geq g$ in $D$ and for every $x\in D$, $$ u(x)\geq p(x)\,φ\left(\frac{G_D(ξψ(p))(x)}{p(x)}\right) $$ where $p=G_Dg$ is the Green function of $g$. The function $φ$ is completely determined by $ψ$ and …
It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional nonlinear systems. In this paper, we restrict ourselves to semilinear infinite-dimensional systems, and show that the exact controllability of the linearized system implies exact controllability of the nonlinear system. The restrictions concerning the nonlinear operator are similar to those that can be found in the literature about the linearized stability analysis…
We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-Δv(x,t) + f(x,v(x,t))= \Un_ω u(x,t)$$ for some nonlinearities $f$ which can produce several non-trivial steady states. One of the usual hypotheses to get global controllability, is to assume that $f(x,v)v\geq 0$. In this case, a stabilisation term $u=γ(x)v_t$ makes any solution converging to zero. The global controllability then follows from a theorem of local controllability and the time reversibility of the equation. In this paper, the nonlinearity $f$ can be more general, so that the solutions of the damped …
Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations
本文研究半线性波动与 Klein-Gordon 方程，在几何控制条件下证明部分解析解及部分爆破解附近的局部能控性，并利用局部能量衰减和全局 Strichartz 估计建立小维数非捕获外域散射解的零能控性。
$L^{p}$-estimates, local well-posedness and controllability for linear and semilinear backward SPDEs
本文研究了线性和半线性反向SPDEs的$L^p$估计、局部适定性与可控性，提出新的Itô公式并解决了半线性方程的局部存在性问题。
In this paper we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
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 …
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $σ$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,m)$, and $f$ is a strictly increasing, convex, continuous function on $[0,\infty)$ with $f(0) = 0$ and $\int_1^\infty 1/f < \infty$. Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as b…
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.
Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation
该研究针对带Dirichlet边界的拟线性多孔介质方程，证明当初值与稳态解的W^{1,n}_0(Ω)∩C(Ω̄)范数差足够小时，方程在有限时间内具备局部精确能控性，拓展了抛物方程能控性理论结果。
We obtain an upper bound on the initial blow-up of nonnegative solutions of second order semilinear parabolic inequalities when a superlinear exponent in the inequalities is not too large.
We consider a $n \times n$ nonlinear reaction-diffusion system posed on a smooth bounded domain $Ω$ of $\mathbb{R}^N$. This system models reversible chemical reactions. We act on the system through $m$ controls ($1 \leq m < n$), localized in some arbitrary nonempty open subset $ω$ of the domain $Ω$. We prove the local exact controllability to nonnegative (constant) stationary states in any time $T >0$. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space $L^\infty(Ω)^n$. The …
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a uniformly local Lebesgue space of a semilinear heat equation \[ \begin{cases} \partial_tu=Δu+f(u) & \textrm{in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x) & \textrm{in}\ \mathbb{R}^N \end{cases} \] under mild assumptions on $f$. A relationship between a growth of $f$ and an integrability of $u_0$ is studied in detail. Our existence theorem gives a sharp integrab…
This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.
Existence of a separating norm for the semilinear heat equation
本文针对Fujita超临界半线性热方程，证明了存在一个分离范数，其小性保证全局解存在，大性导致有限时间爆破，且该范数在非负数据上唯一。
We consider the semilinear heat equation posed on a smooth bounded domain $Ω$ of $\mathbb{R}^{N}$ with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset $ω$ of $Ω$. The goal of this paper is to prove the uniform large time global null-controllability for semilinearities $f(s) = \pm |s| \log^α(2+|s|)$ where $α\in [3/2,2)$ which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global …
Given $Ω$ a bounded open subset of $\mathbb{R}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-Δ\,u\,=\,\frac{f}{u^β}$ in $H^1_{loc}(Ω)$, under zero Dirichlet boundary conditions. For $β>0$ and $f\in L^1(Ω)$, we prove that the solution is unique.
A semilinear heat equation $u_{t}=Δu+f(u)$ with nonnegative initial data in a subset of $L^{1}(Ω)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $Ω\subseteq \R^{n}$. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is applied to the model case $f(s)=s^{p}$, $ϕ\in L^{q}(Ω)$: new sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical range of…
On the exact boundary controllability of semilinear wave equations
本文针对有界区域上的半线性波动方程，在非线性项满足对数型增长条件下，分别用 Schauder 与 Banach 不动点定理证明一致能控性并构造收敛的状态–控制对。
Optimal Control and Approximate controllability of fractional semilinear differential inclusion involving $ψ$- Hilfer fractional derivatives
本文研究含 $ψ$-Hilfer 分数阶导数的半线性微分包含，利用紧半群、线性系统近似能控性及对偶映射构造控制，克服自反 Banach 空间中的凸性难题，并给出最优控制与示例验证。
We consider the heat equation with dynamic bounary conditions involving gradient terms in a bounded domain. In this paper we study the cost of approximate controllability for this equation. Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.
In this paper we consider a semilinear parabolic equation with nonlinear and nonlocal boundary condition and nonnegative initial datum. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.
In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R^N . As an application, we then show the exactly null controllability for this semilinear heat equation in R^N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under considerat…
Nonexistence results for the semilinear wave equation on graphs
该研究针对加权图上带位势的半线性波动方程，推导了全局时间解不存在的充分条件，覆盖非负解与变号解，且变号解的证明运用了该类结果的全新技术。
A New proof of Liouville type theorems for a class of semilinear elliptic equations
本文研究欧氏空间或具非负Ricci曲率的闭流形上的典型半线性椭圆方程，利用不变张量法构造的关键积分恒等式结合积分估计，重新证明了若干经典Liouville定理。
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.
We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-Δu\leq v^λ\] \[0\leq v_t-Δv\leq u^σ\] in $Ω\times (0,1)$, where $λ$ and $σ$ are nonnegative constants and $Ω$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $λ$ and $σ$ such that solutions of this system satisfy pointwise bounds in compact subsets of $Ω$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.
In this paper, we consider systems of semilinear elliptic equations \displaystyle -Δ_{\mathbb{H}^{N}}u=|v|^{p-1}v, \displaystyle -Δ_{\mathbb{H}^{N}}v=|u|^{q-1}u, in the whole of Hyperbolic space $\mathbb{H}^{N}$. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space $\mathbb{R}^N$, we prove that there exists a nonnegative nontrivial solution of problem.
In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.
Nonexistence results for semilinear elliptic equations on weighted graphs
本文研究无限图上带位势的半线性椭圆不等式，在距离的 Laplacian 上界与加权球体积增长条件下，证明非负非平凡解不存在，并验证条件最优。
We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{μt}u^p$ ($μ\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a certain assumption on $μ$ and $p$, related to the bottom of the spectrum of $-Δ$ in $\mathbb H^n$, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters $μ$ and $p$ satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when …
We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.
Threshold property of a singular stationary solution for semilinear heat equations with exponential growth
该文研究高维半线性热方程柯西问题，构造正径向奇异定态解并证明其为阈值：初值低于该解时存在全局解，高于时无满足相应条件的局部解。
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
One proves that the linear and semilinear stochastic parabolic equations with a multiplicative noise with a finite number of modes are exactly null controllable.
In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.
We consider a parabolic equation of the form u_t=Δu +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions, each globally defined and nonnegative bounded solution u converges to a single steady state.
The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=Δu+V(x) |u|^{p-1}u$ in $Ω\times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data $u(x,0)=M\vf (x)$ as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.
In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity …
We study a time-fractional semilinear heat equation $$\partial^α_t u -Δu = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^α$ denotes the Caputo derivative of order $α\in (0,1)$. Since the space $L^1(\mathbb{R}^N)$ is scale critical with $p=1+2/N$, this type of equation is known as a doubly critical problem. It is known that the usual doubly critical equation $\partial_t u-Δu=u^p$ does not have nonnegative global-in-time solutions, while the time-fractional problem does. Moreover, there exists a singular initial …
We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss monotonicity, convexity, decay estimates and triviality of ancient and eternal solutions.
Energy estimate up to the boundary for stable solutions to semilinear elliptic problems
本文针对变系数半线性椭圆方程，建立了稳定解到边界的普适能量估计，仅需 Lipschitz 系数，并推广至凸非线性情形的 $C^{1,1}$ 域，结果对拉普拉斯亦为新。
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.
We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} …
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