Searcharxiv⌕ Search

arXiv subjects

Martina Magliocca

Publications and source records attributed to Martina Magliocca.

16 recordsLinked to original sources
arXiv2026

Remarks on some quasi-linear fourth order parabolic equations arising in mathematical physics

This note proves existence and uniqueness of strong solutions to a broad class of fourth-order quasi-linear parabolic equations in the class of Wiener spaces. It improves upon previous results in the literature, devoted to some special cases falling within the general framework developed here, in which uniqueness held true in a smaller class than the space where existence was proven. Gain of analyticity, as well as generalisations to higher-order equations and to propagation of higher regularities, are also discussed.

math.AP↗
arXiv2026

A new asymptotic model of multilayer tumor growth

In this paper we study the growth of a tumor colony of multilayer type and focus on how the tumor grows from a near flat (when compared to the length of the tumor as, for instance, in the case of a bone tumor in a femur) initial colony. In particular we derive and study a new weakly nonlinear asymptotic model of multilayer tumor growth. The model takes the form of a nonlinear and nonlocal high order system of PDEs. Finally, motivated by the possibility of a finite time collision of the interfaces, we study the well-posedness of this system.

math.AP↗
arXiv2025

Quasi-linear parabolic equations having a superlinear gradient term which depends on the solution

In this paper, we study existence and regularity for solutions to parabolic equations having a superlinear lower order term depending on both the solution and its gradient. Two different situations are analyzed. On the one hand, we assume that the initial datum belongs to an Orlicz space of exponentially summable functions. On the other, data in an appropriate Lebesgue space satisfying a smallness condition are considered. Our results are coherent with those of previous papers in similar frameworks.

math.AP↗
arXiv2025

A nonlocal equation describing tumor growth

Cancer is a very complex phenomenon that involves many different scales and situations. In this paper we consider a free boundary problem describing the evolution of a tumor colony and we derive a new asymptotic model for tumor growth. We focus on the case of a single phase tumor colony taking into account chemotactic effects in an early stage where there is no necrotic inner region. Thus, our model is valid for the case of multilayer avascular tumors with very little access to both nutrients and inhibitors or the case where the amount of nutrients and inhibitors is very similar to the amount consumed by the multilayer tumor cells. Our model takes the form of a single nonlocal and nonlinear partial differential equation. Besides deriving the model, we also prove a well-posedness result.

math.AP↗
arXiv2025

New functional inequalities with applications to the arctan-fast diffusion equation

In this paper, we prove a couple of new nonlinear functional inequalities of Sobolev type akin to the logarithmic Sobolev inequality. In particular, one of the inequalities reads $$ \int_{\mathbb{S}^1}\arctan\left(\frac{\partial_x u}{u}\right)\partial_xu \,dx\geq \arctan\left(\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}\right)\|u(t)\|_{\dot{W}^{1,1}(\mathbb{S}^1)}. $$ Then, these inequalities are used in the study of the nonlinear \emph{arctan}-fast diffusion equation $$ \partial_t u-\partial_x\arctan\left(\frac{\partial_x u}{u}\right)=0. $$ For this highly nonlinear PDE we establish a number of well-posedness results and qualitative properties.

math.AP↗
arXiv2025

Traveling Motility of Actin Lamellar Fragments Under spontaneous symmetry breaking

Cell motility is connected to the spontaneous symmetry breaking of a circular shape. In https://doi.org/10.1103/PhysRevLett.110.078102, Blanch-Mercader and Casademunt perfomed a nonlinear analysis of the minimal model proposed by Callan and Jones https://doi.org/10.1103/PhysRevLett.100.258106 and numerically conjectured the existence of traveling solutions once that symmetry is broken. In this work, we prove analytically that conjecture by means of nonlinear bifurcation techniques.

math.AP↗
arXiv2023

Global existence for certain fourth order evolution equations

In this paper we establish three global in time results for two fourth order nonlinear parabolic equations. The first of such equations involves the Hessian and appears in epitaxial growth. For such equation we give conditions ensuring the global existence of solution. For certain regime of the parameters, our size condition involves the norm in a critical space with respect to the scaling of the equation and improves previous existing results in the literature for this equation. The second of the equations under study is a thin film equation with a porous medium nonlinearity. For this equation we establish conditions leading to the global existence of solution.

math.AP↗
arXiv2022

Traveling wave solution for a coupled incompressible Darcy's free boundary problem with surface tension

We study an incompressible Darcy's free boundary problem, recently introduced in [22]. Our goal is to prove the existence of non-trivial traveling wave solutions and thus validate the interest of this model to describe cell motility. The model equations include a convection diffusion equation for the polarity marker concentration and the incompressible Darcy's equation. The mathematical novelty of this problem is the nonlinear destabilizing term in the boundary condition that describes the active character of the cell cytoskeleton. We first study the linear stability of this problem and we show that, above a well precise threshold, the disk becomes linearly unstable. By using two different approaches we prove existence of traveling wave solutions, which describes persistent motion of a biological cell. One is explicit, by construction. The other is established implicitly, as the one bifurcating from stationary solution.

math.AP↗
arXiv2021

On some parabolic equations involving superlinear singular gradient terms

In this paper we prove existence of nonnegative solutions to parabolic Cauchy-Dirichlet problems with superlinear gradient terms which are possibly singular. The model equation is \[ u_t - Δ_pu=g(u)|\nabla u|^q+h(u)f(t,x)\qquad \text{in }(0,T)\timesΩ, \] where $Ω$ is an open bounded subset of $\mathbb{R}^N$ with $N>2$, $0<T<+\infty$, $1<p<N$, and $q<p$ is superlinear. The functions $g,\,h$ are continuous and possibly satisfying $g(0) = +\infty$ and/or $h(0)= +\infty$, with different rates. Finally, $f$ is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of $q$, the regularity of the initial datum and the forcing term, and the decay rates of $g,\,h$ at infinity.

math.AP↗
arXiv2019

Regularizing effects concerning elliptic equations with a superlinear gradient term

We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as $g(u)|\nabla u|^q$, where $1<q<2$ and $g(s)$ is a continuous function. Data belong to $L^m(Ω)$ with $1\le m <\frac{N}{2}$ as well as measure data instead of $L^1$-data, so that unbounded solutions are expected. Our aim is, given $1\le m<\frac N2$ and $1<q<2$, to find the suitable behaviour of $g$ close to infinity which leads to existence for our problem. We show that the presence of $g$ has a regularizing effect in the existence and summability of the solution. Moreover, our results adjust with continuity with known results when either $g(s)$ is constant or $q=2$.

math.AP↗
arXiv2018

Global existence and decay to equilibrium for some crystal surface models

In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations $$ \partial_t u=Δe^{-Δu}, $$ $$ \partial_t u=-u^2Δ^2(u^3). $$ These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97, 281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.

math.AP↗
arXiv2018

Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

We want to analyse both regularizing effect and long, short time decay concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*} \begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla u)=γ|\nabla u|^q & \text{in}\,\,Q_T,\\ u=0 &\text{on}\,\,(0,T)\times\partialΩ,\\ u(0,x)=u_0(x) &\text{in}\,\, Ω. \end{array} \end{cases} \end{equation*} We assume that $A(t,x)$ is a coercive, bounded and measurable matrix, the growth rate $q$ of the gradient term is superlinear but still subnatural, $γ>0$, the initial datum $u_0$ is an unbounded function belonging to a well precise Lebesgue space $L^σ(Ω)$ for $σ=σ(q,p,N)$.

math.AP↗
arXiv2017

Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth

In this paper we deal with uniqueness of solutions to the following problem \[ \begin{cases} \begin{split} & u_t-Δ_p u=H(t,x,\nabla u) &\quad \text{in}\quad Q_T,\\ & u (t,x) =0 &\quad \text{on}\quad(0,T)\times \partial Ω,\\ & u(0,x)=u_0(x) &\quad \displaystyle\text{in }\quad Ω\end{split} \end{cases} \] where $Q_T=(0,T)\times Ω$ is the parabolic cylinder, $Ω$ is an open subset of $\mathbb{R}^N$, $N\ge2$, $1<p<N$, and the right hand side $\displaystyle H(t,x,ξ):(0,T)\timesΩ\times \mathbb{R}^N\to \mathbb{R}$ exhibits a superlinear growth with respect to the gradient term.

math.AP↗
arXiv2017

Local and global time decay for parabolic equations with super linear first order terms

We study a class of parabolic equations having first order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.

math.AP↗
arXiv2017

Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term

We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following: \[ \begin{cases}\begin{split} & u_t-\text{div}(A(t,x)\nabla u|\nabla u|^{p-2})=γ|\nabla u|^q+f(t,x) &\qquad\text{in } Q_T,\\ & u=0 &\qquad\text{on }(0,T)\times \partial Ω,\\ & u(0,x)=u_0(x) &\qquad\text{in } Ω, \end{split}\end{cases} \] where $Q_T=(0,T)\times Ω$, $Ω$ is a bounded domain of $\mathrm{R}^N$, $N\ge 2$, $1<p<N$, the matrix $A(t,x)$ is coercive and with measurable bounded coefficients, the r.h.s. growth rate satisfies the superlinearity condition \[ \max\left\{\frac{p}{2},\frac{p(N+1)-N}{N+2}\right\}<q<p \] and the initial datum $u_0$ is an unbounded function belonging to a suitable Lebesgue space $L^σ(Ω)$. We point out that, once we have fixed $q$, there exists a link between this growth rate and exponent $σ=σ(q,N,p)$ which allows one to have (or not) an existence result. Moreover, the value of $q$ deeply influences the notion of solution we can ask for. The sublinear growth case with \[ 0<q\le\frac{p}{2} \] is dealt at the end of the paper for what concerns small value of $p$, namely $1<p<2$.

math.AP↗