Kirchhoff systems with nonlinear source and boundary damping terms
Giuseppina Autuori Patrizia Pucci
In this paper we treat the question of the non--existence of global solutions, or their long time behavior, of nonlinear hyperbolic Kirchhoff systems. The main $p$--Kirchhoff operator may be affected by a perturbation which behaves like $|u|^{p-2} u$ and the systems also involve an external force $f$ and a nonlinear boundary damping $Q$. When $p=2$, we consider some problems involving a higher order dissipation term, under dynamic boundary conditions. For them we give criteria in order that $ || u(t,\cdot) ||_q\to\infty$ as $t \to\infty$ along any global solution $u=u(t,x)$, where $q$ is a parameter related to the growth of $f$ in $u$. Special subcases of $f$ and $Q$, interesting in applications, are presented in Sections 4, 5 and 6.
keywords: blow up. nonlinear source and boundary damping terms Kirchhoff systems non continuation
Entire solutions of singular elliptic inequalities on complete manifolds
Patrizia Pucci Marco Rigoli
We present some qualitative properties for solutions of singular quasilinear elliptic differential inequalities on complete Riemannian manifolds, such as the validity of the weak maximum principle at infinity, and non--existence results.
keywords: Quasilinear singular elliptic inequalities on manifolds.
On an initial value problem modeling evolution and selection in living systems
Patrizia Pucci Maria Cesarina Salvatori
This paper is devoted to the qualitative analysis of a new broad class of nonlinear initial value problems that model evolution and selection in living systems derived by the mathematical tools of the kinetic theory of active particles. The paper is divided into two parts. The first shows how to obtain the nonlinear equations with proliferative/distructive nonlinear terms. The latter presents a detailed analysis of the related initial value problem. In particular, it is proved that the corresponding initial value problem admits a unique non--negative maximal solution. However, the solution cannot be in general global in time, due to the possibility of blow--up. The blow--up occurs when the biological life system is globally proliferative, see Theorem 3.3.
keywords: nonlinearity active particles multi--scale modeling kinetic theory complexity in biology living systems nonlinear interactions. Population dynamics
Nonexistence for $p$--Laplace equations with singular weights
Patrizia Pucci Raffaella Servadei
Aim of this paper is to give some nonexistence results of nontrivial solutions for the following quasilinear elliptic equations with singular weights in $R^n\setminus \{0\}$

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{a-2}u+\lambda | u|^{q-2}u+h(|x|)f(u) = 0 $ and

$ \Delta_p u+\mu|x|^{-\alpha}| u|^{p^*_\alpha-2}u+\lambda | u|^{q-2}u+h(|x|)f(u)= 0, $

where $1 < p < n$, $\alpha \in [0,p]$, $a \in [p,p^*_\alpha]$, $p_\alpha^*= p(n-\alpha)/(n-p)$, $\lambda, \mu \in R$ and $q \ge 1$, while $h: R^+ \to R^+_0$ and $f: R\to R$ are given continuous functions. The main tool for deriving nonexistence theorems for the equations is a Pohozaev--type identity. We first show that such identity holds true for weak solutions $u$ in $H^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the first equation and for weak solutions $u$ in $D^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the second equation. Then, under a suitable growth condition on $f$, we prove that every weak solution $u$ has the required regularity, so that the Pohozaev--type identity can be applied. From this identity we derive some nonexistence results, improving several theorems already appeared in the literature. In particular, we discuss the case when $h$ and $f$ are pure powers.

keywords: Hardy–Sobolev embeddings. Quasilinear singular elliptic equations singular weights
Nonlocal Schrödinger-Kirchhoff equations with external magnetic field
Mingqi Xiang Patrizia Pucci Marco Squassina Binlin Zhang
The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case
$\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in $\mathbb{R}^N$},\end{equation*}$
$s∈ (0,1)$
$a∈ \mathbb{R}^+_0$
$b∈ \mathbb{R}^+_0$
is a magnetic potential,
$V:\mathbb{R}^N\to \mathbb{R}^+$
is an electric potential,
$(-Δ )_A^s$
is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.
keywords: Schrödinger-Kirchhoff equation fractional magnetic operators.
Entire solutions of nonlocal elasticity models for composite materials
Giuseppina Autuori Patrizia Pucci

Many structural materials, which are preferred for the developing of advanced constructions, are inhomogeneous ones. Composite materials have complex internal structure and properties, which make them to be more effectual in the solution of special problems required for civil and environmental engineering. As a consequence of this internal heterogeneity, they exhibit complex mechanical properties. In this work, the analysis of some features of the behavior of composite materials under different loading conditions is carried out. The dependence of nonlinear elastic response of composite materials on loading conditions is studied. Several approaches to model elastic nonlinearity such as different stiffness for particular type of loadings and nonlinear shear stress–strain relations are considered. Instead of a set of constant anisotropy coefficients, the anisotropy functions are introduced. Eventually, the combined constitutive relations are proposed to describe simultaneously two types of physical nonlinearities. The first characterizes the nonlinearity of shear stress–strain dependency and the latter determines the stress state susceptibility of material properties. Quite satisfactory correlation between the theoretical dependencies and the results of experimental studies is demonstrated, as described in [2,3] as well as in the references therein.

keywords: Existence theorems entire solutions fractional elliptic operators nonlocal elasticity models composite materials
Existence of radial solutions for the $p$-Laplacian elliptic equations with weights
Elisa Calzolari Roberta Filippucci Patrizia Pucci
Using the definition of solution and the qualitative properties established in the recent paper [17], some existence results are obtained both for crossing radial solutions and for positive or compactly supported radial ground states in $\mathbb R^n$ of quasilinear singular or degenerate elliptic equations with weights and with non--linearities which can be possibly singular at $x=0$ and $u=0$, respectively. The technique used is based on the papers [1] and [12]. Furthermore we obtain a non--existence theorem for radial ground states using a technique of Ni and Serrin [13].
keywords: weight functions. Ground states p-Laplacian operator
Dead cores and bursts for p-Laplacian elliptic equations with weights
Elisa Calzolari Roberta Filippucci Patrizia Pucci
In this paper we consider p-Laplace elliptic equations with weights on domains of $\mathbb{R}^n$, which include several prototypes, and we show that there exist a dead core solution having a burst within the core. This result is obtained by using an existence theorem for ground states having compact support, proved in [4] by the authors, together with qualitative properties and an existence theorem for dead core solutions contained in a recent work of Pucci and Serrin, see [10].
keywords: p-Laplacian operator weight functions. Dead cores
A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian
Patrizia Pucci Mingqi Xiang Binlin Zhang

In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.

keywords: Integro-differential operators anomalous diffusion models sub-differential approach large-time behavior

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