arXiv subjects

Daniel Devine

Publications and source records attributed to Daniel Devine.

5 recordsLinked to original sources

arXiv2024

Stability of positive radial steady states for the parabolic H\'enon-Lane-Emden system

When it comes to the nonlinear heat equation $u_t - \Delta u = u^p$, 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 H\'enon-Lane-Emden system $$\left\{ \begin{aligned} u_t - \Delta u &= |x|^k v^p &\mbox{ in } \mathbb R^n \times (0,\infty),\\ v_t - \Delta v &= |x|^l u^q &\mbox{ in } \mathbb R^n \times (0,\infty), \end{aligned} \right.$$ 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 H\'enon-Lane-Emden system in $\mathbb R^n$.

math.AP↗

arXiv2023

Existence of Positive Radial Solutions of General Quasilinear Elliptic Systems

Let $\Omega\subset\mathbb R^{n}\ (n\geq2)$ be either an open ball $B_R$ centred at the origin or the whole space. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form \begin{equation*} \left\{ \begin{aligned} \Delta_{p} u&=f_1(|x|)g_1(v)|\nabla u|^{\alpha} &&\quad\mbox{ in } \Omega, \\ \Delta_{p} v&=f_2(|x|)g_2(v)h(|\nabla u|) &&\quad\mbox{ in } \Omega, \end{aligned} \right. \end{equation*} where $\alpha\geq 0$, $\Delta_{p}$ is the $p$-Laplace operator, $p>1$, and for $i,j=1,2$ we assume $f_i,g_j,h$ are continuous, non-negative and non-decreasing functions. For functions $g_j$ which grow polynomially, we prove sharp conditions for the existence of positive radial solutions which blow up at $\partial B_{R}$, and for the existence of global solutions.

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} \Delta_p u &= c_1|x|^{m_1} \cdot g_1(v) \cdot |\nabla u|^{\alpha} &\quad\mbox{ in } \mathbb R^n,\\ \Delta_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 $\Delta_p$ denotes the $p$-Laplace operator, $p>1$, $n\geq 2$, $c_1,c_2>0$ and $m_1, m_2, \alpha \geq 0$. For a general class of functions $g_j$ which grow polynomially, we show that every non-constant positive radial solution $(u,v)$ asymptotically approaches $(u_0,v_0) = (C_\lambda |x|^\lambda, C_\mu |x|^\mu)$ for some parameters $\lambda,\mu, C_\lambda, C_\mu>0$. 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↗

arXiv2022

Blow-up radial solutions for elliptic systems with monotonic non-linearities

We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left\{ \begin{aligned} \Delta u&=g(|x|,v(x)) &&\quad\mbox{in}\ \Omega, \\ \Delta v&=f(|x|,|\nabla u(x)|) &&\quad\mbox{in}\ \Omega, \end{aligned} \right. \end{equation*} where $\Omega \subset \mathbb{R}^N$ is either a ball centered at the origin or the whole space $\mathbb{R}^N$, and $f,g\in C^{1}([0,\infty)\times [0,\infty))$, are non-negative, and increasing. Firstly, we study the existence of positive radial solutions in the case when the system is posed in a ball corresponding to their behaviour at the boundary. Next, we discuss the existence of positive radial solutions in case when $g(|x|,v(x)) = |x|^{a} v^p$ and $f(|x|, |\nabla u (x)|) = |x|^{b} h(|\nabla u|)$. Finally, we take $h(t) = t^s$, $s> 1$, $\Omega = \mathbb{R}^N$ and by the use of dynamical system techniques we are able to describe the behaviour at infinity of such positive radial solutions.

math.AP↗

arXiv2022

Existence and boundary behaviour of radial solutions for weighted elliptic systems with gradient terms

We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left\{ \begin{aligned} \Delta u&=|x|^{a}v^{p} &&\quad\mbox{ in } \Omega, \\ \Delta v&=|x|^{b}v^{q}f(|\nabla u|) &&\quad\mbox{ in } \Omega, \end{aligned} \right. \end{equation*} where $\Omega \subset \bR^N$ is either a ball centered at the origin or the whole space $\bR^N$, $a$, $b$, $p$, $q> 0$, and $f \in C^1[0, \infty)$ is an increasing function such that $f(t)> 0$ for all $t> 0$. Firstly, we study the existence of positive radial solutions in case when the system is posed in a ball corresponding to their behaviour at the boundary. Next, we take $f(t) = t^s$, $s> 1$, $\Omega = \bR^N$ and by the use of dynamical system techniques we are able to describe the behaviour at infinity for such positive radial solutions.

math.AP↗