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Philippe Laurençot

Publications and source records attributed to Philippe Laurençot.

At least 19 recordsLinked to original sources
arXiv2026

A Sharp Mass Threshold for Boundedness in a Critical Quasilinear Chemotaxis Model With Indirect Signal Production: The Radially Symmetric Case

A critical quasilinear chemotaxis model with indirect signal production in space dimensions higher than two is studied under radial symmetry. Exploiting the variational structure of the system, the existence of a sharp mass threshold M\textsubscr{c} > 0 is established: all solutions with mass below M\textsubscr{c} are global and bounded, while those with mass above M\textsubscr{c} and sufficiently negative energy are global but blow up in infinite time. Moreover, the value of the critical mass M\textsubscr{c} is related to the best constant of a variant of a Hardy-Littlewood-Sobolev inequality.

math.AP↗
arXiv2026

A sharpened Carlson's integral inequality

A sharpened version of Carlson's integral inequality is established, which features a remainder term involving a relative distance to the set of extremal functions. A similar approach leads to an alternative derivation of Carlson's and Landau's discrete inequalities and provides another characterization of the optimal constant in Carlson's inequality for finite sums.

math.CA↗
arXiv2026

Convergence to self-similarity for a degenerate parabolic equation with fast-growing spatially-dependent absorption

The large time behavior of non-negative solutions to the absorption-diffusion equation $\partial$\textsubscr{t} u = $Δ$ u\textsuperscript{m} - |x|\textsuperscript{$σ$ }u\textsuperscript{m} in (0,$\infty$) x \textsuperscript{N}, with m > 1 and $σ$> $σ$\textsubscr{0}\,:= N (m-1)/(m+1) is identified. It is shown that all solutions approach a unique stationary solution in self-similar variables, which also provides a universal upper bound (friendly giant ), strongly contrasting to the standard case $σ$ = 0. On the one hand, the convergence proof exploits the variational structure of the equation and a suitable Caffarelli-Kohn-Nirenberg inequality, along with the B{é}nilan-Crandall homogeneity regularizing effect. On the other hand, the detailed study of the stationary problem combines elliptic estimates, Moser iteration and techniques from ordinary differential equations.

math.AP↗
arXiv2025

Dynamics of the reversible Gray-Scott model and convergence to its irreversible limit

Well-posedness of a reversible variant of the Gray-Scott model is shown, along with the convergence of each trajectory to one of the two spatially homogeneous steady states. The principle of linearized stability provides the local attractivity at an exponential rate of the stable steady state, while the long-term limit is identified with the help of center manifold theory. Finally, convergence to the classical Gray-Scott model is proved for an appropriate choice of parameters.

math.AP↗
arXiv2025

Sharp non-existence threshold for a parabolic Hardy-H{é}non equation with quasilinear diffusion

Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{é}non equation $$ \partial\_tu=Δu^m+|x|^σu^p, \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, $$ with $m>0$, $σ>0$ and $p>\max\{1,m\}$, are identified. Assuming that the initial condition satisfies $$ u\_0\in L^{\infty}(\mathbb{R}^N), \quad \lim\limits\_{|x|\to\infty}|x|^γu\_0(x)=L\in(0,\infty), \quad u\_0\geq0, $$ it is shown that non-existence of solution occurs for $$ γ<\frac{σ+2}{p-m} - \frac{2\max{\{p-p\_G,0\}}}{(p-1)(p-m)} $$ with $$ p\_G:=1+\frac{σ(1-m)}{2}. $$ The above threshold for non-existence is optimal, in view of the existence of self-similar solutions for the limiting value of $γ$.

math.AP↗
arXiv2025

Global Solutions to the Discrete Nonlinear Breakage Equations without Mass Transfer

Global existence of mild solutions to the discrete collisional breakage equations is established for a broad class of collision kernels, without imposing any growth assumptions. In addition, classical solutions are constructed, and uniqueness is proved for an appropriate class of kinetic coefficients and initial data. The large time behavior of solutions is also discussed, and numerical simulations are presented to support the theoretical results.

math.CA↗
arXiv2025

A parabolic Hardy-Hénon equation with quasilinear degenerate diffusion

Local and global well-posedness, along with finite time blow-up, are investigated for the following Hardy-Hénon equation involving a quasilinear degenerate diffusion and a space-dependent superlinear source featuring a singular potential $$\partial_t u=Δu^m+|x|^σu^p,\quad t>0,\ x\in\mathbb{R}^N,$$ when $m>1$, $p>1$ and $σ\in \big(\max\{-2,-N\},0 \big)$. While the superlinear source induces finite time blow-up when $σ=0$, whatever the value of $p>1$, at least for sufficiently large initial conditions, a striking effect of the singular potential $|x|^σ$ is the prevention of finite time blow-up for suitably small values of $p$, namely, $1 p_G$, is obtained by employing the Caffarelli-Kohn-Nirenberg inequalities. Another interesting feature is that uniqueness and comparison principle hold true for generic non-negative initial conditions when $p>p_G$, but their validity is restricted to initial conditions which are positive in a neighborhood of $x=0$ when $p\in (1,p_G)$, a range in which non-uniqueness holds true without this positivity condition. Finite time blow-up of any non-trivial, non-negative solution is established when $p_G p_F$. Optimal temporal growth rates are also derived for global solutions when $p\in (1,p_G]$. All the results are sharp with respect to the exponents $(m,p,σ)$ and conditions on $u_0$.

math.AP↗
arXiv2024

Global boundedness induced by asymptotically non-degenerate motility in a fully parabolic chemotaxis model with local sensing

A fully parabolic chemotaxis model of Keller-Segel type with local sensing is considered. The system features a signal-dependent asymptotically non-degenerate motility function, which accounts for a repulsion-dominated chemotaxis. Global boundedness of classical solutions is proved for an initial Neumann boundary value problem of the system in any space dimension. In addition, stabilization towards the homogeneous steady state is shown, provided that the motility is monotone non-decreasing. The key steps of the proof consist of the introduction of several auxiliary functions and a refined comparison argument, along with uniform-in-time estimates for a functional involving nonlinear coupling between the unknowns and auxiliary functions.

math.AP↗
arXiv2024

Singular limit of a chemotaxis model with indirect signal production and phenotype switching

Convergence of solutions to a partially diffusive chemotaxis system with indirect signal production and phenotype switching is shown in a two-dimensional setting when the switching rate increases to infinity, thereby providing a rigorous justification of formal computations performed in the literature. The expected limit system being the classical parabolic-parabolic Keller-Segel system, the obtained convergence is restricted to a finite time interval for general initial conditions but valid for arbitrary bounded time intervals when the mass of the initial condition is appropriately small. Furthermore, if the solution to the limit system blows up in finite time, then neither of the two phenotypes in the partially diffusive system can be uniformly bounded with respect to the $L_2$-norm beyond that time.

math.AP↗
arXiv2024

A Hardy-H{é}non equation in $\mathbb{R}^N$ with sublinear absorption

Consider $m\>1$, $N\ge 1$ and $\max\{-2,-N\}\<σ\<0$. The Hardy-Hénon equation with sublinear absorption\begin(equation*}- Δv(x) - |x|^σv(x) + \frac{1}{m-1} v^{1/m}(x)= 0, \qquad x\in\mathbb{R}^N,\end{equation*}is shown to have at least one solution $v\in H^1(\mathbb{R}^N)\cap L^{(m+1)/m}(\mathbb{R}^N)$, which is non-negative and radially symmetric with a non-increasing profile. In addition, any such solution is compactly supported, bounded and enjoys the better regularity $v\in W^{2,q}(\mathbb{R}^N)$ for $q\in [1,N/|σ|)$. A key ingredient in the proof is a particular case of the celebrated Caffarelli-Kohn-Nirenberg inequalities, for which we obtain the existence of an extremal function which is non-negative, bounded, compactly supported and radially symmetric with a non-increasing profile.A by-product of these results is the existence of compactly supported separate variables solutions to a porous medium equation with a spatially dependent source featuring a singular coefficient.

math.AP↗
arXiv2024

Well-posedness of the growth-coagulation equation with singular kernels

The well-posedness of the growth-coagulation equation is established for coagulation kernels having singularity near the origin and growing atmost linearly at infinity. The existence of weak solutions is shown by means of the method of the characteristics and a weak $L_1$-compactness argument. For the existence result, we also show our gratitude to Banach fixed point theorem and a refined version of the Arzelá-Ascoli theorem. In addition, the continuous dependence of solutions upon the initial data is shown with the help of the DiPerna-Lions theory, Gronwall's inequality and moment estimates. Moreover, the uniqueness of solution follows from the continuous dependence. The results presented in this article extend the contributions made in earlier literature.

math.AP↗
arXiv2024

The Redner-ben-Avraham-Kahng cluster system without growth condition on the kinetic coefficients

Existence of global mild solutions to the infinite dimensional Redner--ben-Avraham--Kahng cluster system is shown without growth or structure condition on the kinetic coefficients, thereby extending previous results in the literature. The key idea is to exploit the dissipative features of the system to derive a control on the tails of the infinite sums involved in the reaction terms. Classical solutions are also constructed for a suitable class of kinetic coefficients and initial conditions.

math.CA↗
arXiv2024

Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles

Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q = 0 \;\;\text{ in }\;\; (0,\infty)\times\mathbb{R}^N,$$ with $m>1$, $q\in(0,1)$ and $σ=2(1-q)/(m-1)$. Looking for solutions of the form $$ u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$ it is shown that, for $m+q>2$, there exists a unique exponent $β_*\in(0,\infty)$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.

math.AP↗
arXiv2024

Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $$\partial_t u - \mathrm{div}(|\nabla u|^{p-2}\nabla u) +|\nabla u|^{q}=0 \qquad {\rm in} \ (0,\infty)\times\mathbb{R}^N$$ are studied when $N\geq1$ and the exponents $(p,q)$ satisfy $p_c=\frac{2N}{N+1}$, $p-1<q<\frac{p}{2}$. Existence and uniqueness of such a solution are established in dimension $N=1$. In dimension $N\geq2$, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as $|x|\to\infty$ is provided.

math.AP↗
arXiv2024

No pattern formation in a quasilinear chemotaxis model with local sensing

Convergence to spatially homogeneous steady states is shown for a chemotaxis model with local sensing and possibly nonlinear diffusion when the intrinsic diffusion rate $ϕ$ dominates the inverse of the chemotactic motility function $γ$, in the sense that $(ϕγ)'\ge 0$. This result encompasses and complies with the analysis and numerical simulations performed in Choi \& Kim (2023). The proof involves two steps: first, a Liapunov functional is constructed when $ϕγ$ is non-decreasing. The convergence proof relies on a detailed study of the dissipation of the Liapunov functional and requires additional technical assumptions on $ϕ$ and $γ$.

math.AP↗
arXiv2024

Global existence and boundedness of solutions to a fully parabolic chemotaxis system with indirect signal production in $\mathbb{R}^4$

Global existence and boundedness of solutions to the Cauchy problem for the four dimensional fully parabolic chemotaxis system with indirect signal production are studied. We prove that solutions with initial mass below $(8π)^2$ exist globally in time. This value $(8π)^2$ is known as the four dimensional threshold value of the initial mass determining whether blow-up of solutions occurs or not. Furthermore, some condition on the initial mass guaranteeing that the solution remains uniformly bounded is also obtained.

math.AP↗
arXiv2024

The continuous collision-induced nonlinear fragmentation equation with non-integrable fragment daughter distributions

Existence, non-existence, and uniqueness of mass-conserving weak solutions to the continuous collision-induced nonlinear fragmentation equations are established for the collision kernels $Φ$ satisfying $Φ(x_1,x_2)={x_1}^{λ_1} {x_2}^{λ_2}+{x_2}^{λ_1} {x_1}^{λ_2}$, $(x_1,x_2)\in(0,\infty)^2$, with ${λ_1} \leq {λ_2}\leq 1$, and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when $1\leλ:={λ_1}+{λ_2}\le2$ with $λ_1\ge k_0$, the parameter $k_0\in(0,1)$ being related to the non-integrability of the fragment daughter distribution. The existence of at least one mass-conserving weak solution is also demonstrated when $2k_0 \le λ< 1$ with $λ_1\ge k_0$ but its maximal existence time is shown to be finite. Uniqueness is also established in both cases. The last result deals with the non-existence of mass-conserving weak solutions, even on a small time interval, for power law fragment daughter distribution when $λ_1<k_0$. It is worth mentioning that the previous literature on the nonlinear fragmentation equation does not treat non-integrable fragment daughter distribution functions.

math.AP↗