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Paschalis Karageorgis

Publications and source records attributed to Paschalis Karageorgis.

15 recordsLinked to original sources
arXiv2024

Stability of positive radial steady states for the parabolic Hénon-Lane-Emden system

When it comes to the nonlinear heat equation $u_t - Δu = u^p$, the stability of positive radial steady states in the supercritical case was established in the classical paper by Gui, Ni and Wang. We extend this result to systems of reaction-diffusion equations by studying the positive radial steady states of the parabolic Hénon-Lane-Emden system $$\left\{ \begin{aligned} u_t - Δu &= |x|^k v^p &\mbox{ in } \mathbb R^n \times (0,\infty),\\ v_t - Δv &= |x|^l u^q &\mbox{ in } \mathbb R^n \times (0,\infty), \end{aligned} \right.$$ where $k,l\geq 0$, $p,q\geq 1$ and $pq>1$. Assume that $(p,q)$ lies either on or above the Joseph-Lundgren critical curve which arose in the work of Chen, Dupaigne and Ghergu. Then all positive radial steady states have the same asymptotic behavior at infinity, and they are all stable solutions of the parabolic Hénon-Lane-Emden system in $\mathbb R^n$.

math.AP↗
arXiv2023

Monotonic convergence of positive radial solutions for general quasilinear elliptic systems

We study the asymptotic behavior of positive radial solutions for quasilinear elliptic systems that have the form \begin{equation*} \left\{ \begin{aligned} Δ_p u &= c_1|x|^{m_1} \cdot g_1(v) \cdot |\nabla u|^α &\quad\mbox{ in } \mathbb R^n,\\ Δ_p v &= c_2|x|^{m_2} \cdot g_2(v) \cdot g_3(|\nabla u|) &\quad\mbox{ in } \mathbb R^n, \end{aligned} \right. \end{equation*} where $Δ_p$ denotes the $p$-Laplace operator, $p>1$, $n\geq 2$, $c_1,c_2>0$ and $m_1, m_2, α\geq 0$. For a general class of functions $g_j$ which grow polynomially, we show that every non-constant positive radial solution $(u,v)$ asymptotically approaches $(u_0,v_0) = (C_λ|x|^λ, C_μ|x|^μ)$ for some parameters $λ,μ, C_λ, C_μ>0$. In fact, the convergence is monotonic in the sense that both $u/u_0$ and $v/v_0$ are decreasing. We also obtain similar results for more general systems.

math.AP↗
arXiv2021

Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_α\ast u^p)u^q$ in $Ω$, where $Ω\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_α$ stands for the Riesz potential of order $α\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $α$ for which positive solutions exist. Our methods are then extended to quasilinear elliptic systems of inequalities.

math.AP↗
arXiv2020

Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

We study the quasilinear elliptic inequality $$ -Δ_m u - \fracμ{|x|^m}u^{m-1} \geq (I_α*u^p)u^q \quad\mbox{ in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, $$ where $p>0$, $q, μ\in \mathbb{R}$, $m>1$ and $I_α$ is the Riesz potential of order $α\in (0,N)$. We obtain necessary and sufficient conditions for the existence of positive solutions.

math.AP↗
arXiv2012

A lower bound for the amplitude of traveling waves of suspension bridges

We obtain a lower bound for the amplitude of nonzero homoclinic traveling wave solutions of the McKenna--Walter suspension bridge model. As a consequence of our lower bound, all nonzero homoclinic traveling waves become unbounded as their speed of propagation goes to zero (in accordance with numerical observations).

math.AP↗
arXiv2012

Dispersion relation for water waves with non-constant vorticity

We derive the dispersion relation for linearized small-amplitude gravity waves for various choices of non-constant vorticity. To the best of our knowledge, this relation is only known explicitly in the case of constant vorticity. We provide a wide range of examples including polynomial, exponential, trigonometric and hyperbolic vorticity functions.

physics.flu-dyn↗
arXiv2007

Supercritical biharmonic equations with power-type nonlinearity

The biharmonic supercritical equation $Δ^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $λ(1+u)^p$ as right-hand-side with an additional eigenvalue parameter $λ>0$ in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the explicitly known radial {\it singular} solution, provided $p\in((n+4)/(n-4),p_c)$, where $p_c\in ((n+4)/(n-4),\infty]$ is a further critical exponent, which was introduced in a recent work by Gazzola and the second author. The third author proved already that these oscillations do not occur in the complementing case, where $p\ge p_c$. Concerning the Dirichlet problem we prove existence of at least one singular solution with corresponding eigenvalue parameter. Moreover, for the extremal solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue problem, we prove smoothness as long as $p\in((n+4)/(n-4),p_c)$.

math.AP↗
arXiv2007

Sharp bounds on 2m/r for static spherical objects

Sharp bounds are obtained, under a variety of assumptions on the eigenvalues of the Einstein tensor, for the ratio of the Hawking mass to the areal radius in static, spherically symmetric space-times.

gr-qc↗
arXiv2007

Stability and intersection properties of solutions to the nonlinear biharmonic equation

We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $Δ^2 ϕ= ϕ^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions are linearly unstable if $p<p_c$ and linearly stable if $p\geq p_c$. Then, we focus on the supercritical case $p\geq p_c$ and we show that the graphs of no two solutions intersect one another.

math.AP↗
arXiv2006

Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as $\d_{tt}u -Δu= f(x,u)$ where $f$ is convex in $u$. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as $\d_t u -Δu= f(x,u)$. Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.

math.AP↗
arXiv2005

Small-data scattering for nonlinear waves with potential and initial data of critical decay

We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp existence results for the Cauchy problem with small initial data; those require the data to decay at a rate greater than or equal to a critical decay rate which depends on the order of the nonlinearity. However, scattering results have appeared only for the supercritical case. In this paper, we extend the scattering results to the critical case and we also allow the presence of a short-range potential.

math.AP↗
arXiv2005

Existence and blow up of small-amplitude nonlinear waves with a sign-changing potential

We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time.

math.AP↗