arXiv subjects

Philippe Laurençot

Publications and source records attributed to Philippe Laurençot.

At least 37 records · Page 2Linked to original sources

arXiv2023

Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles

Existence of specific \emph{eternal solutions} in exponential self-similar form to the following quasilinear diffusion equation with strong absorption$$\partial_t u=Δu^m-|x|^σu^q,$$posed for $(t,x)\in(0,\infty)\times\mathbb{R}^N$, with $m>1$, $q\in(0,1)$ and $σ=σ_c:=2(1-q)/(m-1)$ is proved. Looking for radially symmetric solutions of the form$$u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$we show that there exists a unique exponent $β^*\in(0,\infty)$ for which there exists a one-parameter family $(u_A)_{A>0}$ of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$ satisfying $f_A(0)=A$ and $f_A'(0)=0$. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $σ\in (0,σ_c)$.

math.AP↗

arXiv2023

The discrete collision-induced breakage equation with mass transfer: well-posedness and stationary solutions

The discrete collisional breakage equation, which captures the dynamics of cluster growth when clusters encounter binary collisions with possible matter transfer, is discussed in this article. The existence of global mass-conserving solutions is investigated for the collision kernels $a_{i,j}=A(i^α j^β + i^βj^α)$, $i, j \ge 1$, with $α\in (-\infty,1)$, $β\in [α,1]\cap (0,1]$, and $A>0$ and for a large class of possibly unbounded daughter distribution functions. All algebraic superlinear moments of these solutions are bounded on time intervals $[T,\infty)$ for any $T>0$. The uniqueness issue is further handled under additional restrictions on the initial data. Finally, non-trivial stationary solutions are constructed by a dynamical approach.

math.CA↗

arXiv2023

A nonlocal Gray-Scott model: well-posedness and diffusive limit

Well-posedness in $L_\infty$ of the nonlocal Gray-Scott model is studied for integrable kernels, along with the stability of the semi-trivial spatially homogeneous steady state. In addition, it is shown that the solutions to the nonlocal Gray-Scott system converge to those to the classical Gray-Scott system in the diffusive limit.

math.AP↗

arXiv2023

Large time convergence for a chemotaxis model with degenerate local sensing and consumption

Convergence to a steady state in the long term limit is established for global weak solutions to a chemotaxis model with degenerate local sensing and consumption, when the motility function is C^1-smooth on [0, $\infty$), vanishes at zero, and is positive on (0, $\infty$). A condition excluding that the large time limit is spatially homogeneous is also provided. These results extend previous ones derived for motility functions vanishing algebraically at zero and rely on a completely different approach.

math.AP↗

arXiv2023

Long term spatial homogeneity for a chemotaxis model with local sensing and consumption

Global weak solutions to a chemotaxis model with local sensing and consumption are shown to converge to spatially homogeneous steady states in the large time limit, when the motility is assumed to be positive and $C^1$-smooth on $[0,\infty)$. The result is valid in arbitrary space dimension $n\ge 1$ and extends a previous result which only deals with space dimensions $n\in \{1,2,3\}$.

math.AP↗

arXiv2023

Global bounded classical solutions to a parabolic-elliptic chemotaxis model with local sensing and asymptotically unbounded motility

Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of $L^\infty$-estimates on the second component of the system and is achieved by various comparison arguments.

math.AP↗

arXiv2022

Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems

Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.

math.AP↗

arXiv2022

Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$, $q\in(0,1)$ and $σ>0$, is addressed. Introducing the critical exponent $σ^* := 2(1-q)/(m-1)$ for $m>1$ and $σ_*=\infty$ for $m=1$, extinction in finite time is known to take place for $σ\in [0,σ^*)$ and an alternative proof is provided therein. When $m>1$ and $σ\ge σ^*$, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of $σ$ and showing their sharpness.

math.AP↗

arXiv2022

Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=Δu^m-|x|^σu^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $σ> 2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0 \in L^\infty(\mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t > 0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.

math.AP↗

arXiv2022

Non-existence of nonnegative separate variable solutions to a porous medium equation with spatially dependent nonlinear source

The non-existence of nonnegative compactly supported classical solutions to $$- ΔV(x) - |x|^σV(x) + \frac{V^{1/m}(x)}{m-1} = 0, \qquad x\in\mathbb{R}^N,$$ with $m>1$, $σ>0$, and $N\ge 1$, is proven for $σ$ sufficiently large. More precisely, in dimension $N\geq4$, the optimal lower bound on $σ$ for non-existence is identified, namely $$σ\geqσ_c := \frac{2(m-1)(N-1)}{3m+1},$$ while, in dimensions $N\in\{1,2,3\}$, the lower bound derived on $σ$ improves previous ones already established in the literature. A by-product of this result is the non-existence of nonnegative compactly supported separate variable solutions to a porous equation medium equation with spatially dependent superlinear source.

math.AP↗

arXiv2022

Stationary states to a free boundary transmission problem for an electrostatically actuated plate

A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the deflection of the plate is derived. The latter paves the way for the construction of energy minimizers and also provides the Euler-Lagrange equation satisfied by these minimizers. A by-product is the monotonicity of the electrostatic energy with respect to the deflection.

math.AP↗

arXiv2022

Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing

New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose.

math.AP↗

arXiv2022

Bounded Weak Solutions to a Class of Degenerate Cross-Diffusion Systems

Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence $(\mathcal{E}_n)_{n\ge 2}$ of Liapunov functionals such that $\mathcal{E}_n$ is equivalent to the $L_n$-norm for each $n \ge 2$ and $\mathcal{E}_n^{1/n}$ controls the $L_\infty$-norm in the limit $n\to\infty$. Weak solutions are built by a compactness approach, special care being needed in the construction of the approximation in order to preserve the availability of the above-mentioned Liapunov functionals.

math.AP↗

arXiv2021

Well-posedness of the coagulation-fragmentation equation with size diffusion

Local and global well-posedness of the coagulation-fragmentation equation with size diffusion are investigated. Owing to the semilinear structure of the equation, a semigroup approach is used, building upon generation results previously derived for the linear fragmentation-diffusion operator in suitable weighted $L^1$-spaces.

math.AP↗

arXiv2021

Convergence of energy minimizers of a MEMS model in the reinforced limit

Energy minimizers to a MEMS model with an insulating layer are shown to converge in its reinforced limit to the minimizer of the limiting model as the thickness of the layer tends to zero. The proof relies on the identification of the $Γ$-limit of the energy in this limit.

math.AP↗

arXiv2021

Weak Solutions to the Collision-Induced Breakage Equation with Dominating Coagulation

Existence and uniqueness of weak solutions to the collision-induced breakage and coag-ulation equation are shown when coagulation is the dominant mechanism for small volumes. The collision kernel may feature a stronger singularity for small volumes than the ones considered in previous contributions. In addition, when the collision kernel is locally bounded, the class of fragment daughter distribution functions included in the analysis is broader. Mass-conserving solutions are also constructed when the collision kernel grows at most linearly at infinity and are proved to be unique for initial conditions decaying sufficiently fast at infinity. The existence proofs relies on a weak compactness approach in L 1 .

math.AP↗

arXiv2021

Reinforced limit of a MEMS model with heterogeneous dielectric properties

A MEMS model with an insulating layer is considered and its reinforced limit is derived by means of a Gamma convergence approach when the thickness of the layer tends to zero. The limiting model inherits the dielectric properties of the insulating layer.

math.AP↗

arXiv2021

Existence and non-existence for the collision-induced breakage equation

A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functions, assuming the collision kernel $K$ to be given by $K(x,y)= x^α y^β + x^β y^α$ with $α\le β\le 1$. When $α+ β\in [1,2]$, it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when $α+ β\in [0,1)$ and $α\ge 0$, global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for $α<0$ and a specific daughter distribution function, the non-existence of mass-conserving solutions is also established.

math.AP↗