Searcharxiv⌕ Search

arXiv subjects

Razvan Gabriel Iagar

Publications and source records attributed to Razvan Gabriel Iagar.

63 records · Page 4Linked to original sources
arXiv2015

Large time behavior for a quasilinear diffusion equation with critical gradient absorption

We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption $$\partial\_t u - Δ\_{p}u + |\nabla u|^{q\_*} = 0 \quad \hbox{in} (0,\infty)\times\mathbb{R}^N\ ,$$ for $p\in(2,\infty)$ and $q\_*:=p-N/(N+1)$. We show that the asymptotic profile of compactly supported solutions is given by a source-type self-similar solution of the $p$-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.

math.AP↗
arXiv2014

Large time behavior for the fast diffusion equation with critical absorption

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption $$ \partial_{t}u-Δu^m+u^q=0 \quad \quad \hbox{in} \ (0,\infty)\times\real^N\, $$ with $m_c:=(N-2)_{+}/N < m < 1$ and $q=m+2/N$. Given an initial condition $u_0$ decaying arbitrarily fast at infinity, we show that the asymptotic behavior of the corresponding solution $u$ is given by a Barenblatt profile with a logarithmic scaling, thereby extending a previous result requiring a specific algebraic lower bound on $u_0$. A by-product of our analysis is the derivation of sharp gradient estimates and a universal lower bound, which have their own interest and hold true for general exponents $q > 1$.

math.AP↗
arXiv2013

Rotationally symmetric p-harmonic flows from D^2 to S^2: local well-posedness and blow-up

We study the $p$-harmonic flow from the unit disk $D^2$ to the unit sphere $S^2$ under rotational symmetry. We show that the Dirichlet problem with constant boundary conditions is locally well-posed in the class of classical solutions and we also give a sufficient criterion, in terms of the boundary condition, for the derivative of the solutions to blow-up in finite time.

math.AP↗
arXiv2012

Rotationally symmetric p-harmonic maps from D^2 to S^2

We consider rotationally symmetric $p$-harmonic maps from the unit disk $D^2\subset\real^2$ to the unit sphere $S^2\subset\real^3$, subject to Dirichlet boundary conditions and with $1<p<\infty$. We show that the associated energy functional admits a unique minimizer which is of class $C^{\infty}$ in the interior and $C^1$ up to the boundary. We also show that there exist infinitely many global solutions to the associated Euler-Lagrange equation and we completely characterize them.

math.AP↗
arXiv2012

Eternal solutions to a singular diffusion equation with critical gradient absorption

The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type $u(t,x)=e^{-pβt/(2-p)} f_β(|x|e^{-βt};β)$ is investigated for the singular diffusion equation with critical gradient absorption \partial_{t} u-Δ_{p} u+|\nabla u|^{p/2}=0 \quad \;\;\hbox{in}\;\; (0,\infty)\times\real^N where $2N/(N+1) < p < 2$. Such solutions are shown to exist only if the parameter $β$ ranges in a bounded interval $(0,β_*]$ which is in sharp contrast with well-known singular diffusion equations such as $\partial_{t}ϕ-Δ_{p} ϕ=0$ when $p=2N/(N+1)$ or the porous medium equation $\partial_{t}ϕ-Δϕ^m=0$ when $m=(N-2)/N$. Moreover, the profile $f(r;β)$ decays to zero as $r\to\infty$ in a faster way for $β=β_*$ than for $β\in (0,β_*)$ but the algebraic leading order is the same in both cases. In fact, for large $r$, $f(r;β_*)$ decays as $r^{-p/(2-p)}$ while $f(r;β)$ behaves as $(\log r)^{2/(2-p)} r^{-p/(2-p)}$ when $β\in (0,β_*)$.

math.AP↗
arXiv2011

Positivity, decay, and extinction for a singular diffusion equation with gradient absorption

We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \partial_t u -Δ_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where $N\ge 1$, $p\in(1,2)$, and $q>0$. Based on gradient estimates for the solutions, we classify the behavior of the solutions for large times, obtaining either positivity as $t\to\infty$ for $q>p-N/(N+1)$, optimal decay estimates as $t\to\infty$ for $p/2\le q\le p-N/(N+1)$, or extinction in finite time for $0 < q < p/2$. In addition, we show how the diffusion prevents extinction in finite time in some ranges of exponents where extinction occurs for the non-diffusive Hamilton-Jacobi equation.

math.AP↗
arXiv2010

Asymptotic Behaviour of a Nonlinear Parabolic Equation with Gradient Absorption and Critical Exponent

We study the large-time behaviour of the solutions of the evolution equation involving nonlinear diffusion and gradient absorption, $$ \partial_t u - Δ_p u + |\nabla u|^q=0 . $$ We consider the problem posed for $x\in \real^N$ and t>0 with nonnegative and compactly supported initial data. We take the exponent p>2 which corresponds to slow p-Laplacian diffusion. The main feature of the paper is that the exponent q takes the critical value q=p-1 which leads to interesting asymptotics. This is due to the fact that in this case both the Hamilton-Jacobi term $ |\nabla u|^q$ and the diffusive term $Δ_p u$ have a similar size for large times. The study performed in this paper shows that a delicate asymptotic equilibrium happens, so that the large-time behaviour of the solutions is described by a rescaled version of a suitable self-similar solution of the Hamilton-Jacobi equation $|\nabla W|^{p-1}=W$, with logarithmic time corrections. The asymptotic rescaled profile is a kind of sandpile with a cusp on top, and it is independent of the space dimension.

math.AP↗