Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.37662

Sharp decay rate and asymptotic simplification for a quasilinear heat equation

Maryam Al Hajjar·Ioana Ciotir·Matthias Täufer
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the decay rate for a quasilinear heat equation of the form $ \dot u-Δu = \operatorname{div B}(\nabla u)$. The term on the right-hand-side is interpreted as a nonlinear perturbation of the ordinary heat equation and we will prove an optimal decay rate for the solution. More precisely, under natural conditions on the operator $B \colon \R^d \to \R^d$, we show that for sufficiently regular initial data the solution decays as $t^{-d/4}$ and its gradient as $t^{-d/4 - 1/2}$ for large times, that is at the same rates as for the free heat equation. Furthermore, we prove that the difference between the quasilinear and the linear solution decays strictly faster than the solution of either equation generically does, that is that there is asymptotic simplification. All estimates are completely explicit and rely on variations of Fourier splitting techniques, originally introduced by Schonbek.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maryam Al Hajjar, Ioana Ciotir, Matthias Täufer. 2026-09-29. Sharp decay rate and asymptotic simplification for a quasilinear heat equation. https://arxiv.org/abs/2609.37662

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Optimizers in Sobolev-curl inequalities

We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer $u:\mathbb{R}^3\to \mathbb{R}^3$ which yields a solution to the $p$-curl-curl equation in the critical case $$ \nabla\times (|\nabla\times u|^{p-2}\nabla\times u)= |u|^{p^*-2}u\quad \hbox{in }\mathbb{R}^3.$$ The cases $p=2$ and $p=3/2$ are motivated, respectively, by nonlinear Maxwell equations and zero modes of three-dimensional Dirac operators. The infinite-dimensional kernel of curl and the critical exponent prevent a direct application of standard compactness arguments. Our proof combines local compactness under the nonlinear constraint $\operatorname{div}(|u|^{p^*-2}u)=0$ and a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also establish existence and compactness results in axially symmetric classes, including two types of solutions for $p=3/2$ distinct from the explicit Loss--Yau fields. Finally, the same minimization strategy gives a new proof of compactness modulo translations and dilations for minimizing sequences in the classical critical Sobolev inequality.

math.AP↗
arXiv2026

A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ dρ\propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$ When $2 \le p \le 4$, we show that these measures indeed satisfy a log-Sobolev inequality. When $p> 4$, we show a lower bound for the Hessian of the potential, which implies that the known techniques to show these inequalities cannot apply to the measure $ρ$.

math.AP↗
arXiv2026

On the inhomogeneous discounted Hamilton-Jacobi equations

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} λ(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $λ(x)$. There is a critical value $c_0\in[-\infty,\infty)$ such that \eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $c c_0$. In this case, we determine the basin of the stable solution and investigate the long-time behavior of the solution semigroup associated with \eqref{hjs1}. In particular, we obtain a formula relating the lowest convergence rate of the solution semigroup near a stable solution to a minimizing problem concerning the integral of $λ$ over Mather measures supporting on the $1$-graph of the stable solution. This formula has two direct consequences: 1) it yields a description of their asymptotic behavior as $c$ tends to infinity; 2) it helps to classify the ergodic Mather measures and locate their distribution in the phase space. The second consequence leads to a dynamical necessary condition for $c=c_0$.

math.AP↗