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Mustapha Jazar

Publications and source records attributed to Mustapha Jazar.

10 recordsLinked to original sources
arXiv2016

Existence result for degenerate cross-diffusion system with application to seawater intrusion

In this paper, we study degenerate parabolic system, which are strongly coupled. We prove general existence result, but the uniqueness remains an open question. Our proof of existence is based on a crucial entropy estimate which both control the gradient of the solution and the non-negativity of the solution. Our system are of porous medium type and our method applies to models in seawater intrusion.

math.AP↗
arXiv2014

A priori gradient bounds for fully nonlinear parabolic equations and applications to porous medium models

We prove a priori gradient bounds for classical solutions of the fully nonlinear parabolic equation $$u_{t}=F(D^2u,D u,u,x,t).$$ The domain is the torus {\mathbb{T}}^{d} of dimension $d\ge1$. Up to the price of technicalities, our work can be extended to the case of bounded domains or the case of the whole space ${\mathbb{R}}^d$. Several applications are given, including the standard porous medium equation.

math.AP↗
arXiv2014

Explicit phase diagram for a one-dimensional blister model

In this article, we consider a simple one-dimensional variational model, describing the delamination of thin films under cooling. We characterize the global minimizers, which correspond to films of three possible types: non delaminated, partially delaminated (called blisters), or fully delaminated. Two parameters play an important role: the length of the film and the cooling parameter. In the phase plane of those two parameters, we classify all the minimizers. As a consequence of our analysis, we identify explicitly the smallest possible blisters for this model.

math-ph↗
arXiv2008

Pseudo-radial solutions of semi-linear elliptic equations on symmetric domains

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $Δu = \pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains.

math.AP↗
arXiv2008

Reduced measures associated to parabolic problems

We study the existence and the properties of the reduced measures for the parabolic equations $\partial_tu-Δu+g(u)=0$ in $Ω\times (0,\infty)$ subject to the conditions ($P$): $u=0$ on $\partialΩ\times (0,\infty)$, $u(x,0)=μ$ and ($P'$): $u=μ'$ on $\partialΩ\times (0,\infty)$, $u(x,0)=0$ where $μ$ and $μ'$ are positive Radon measures and $g$ a continuous nondecreasing function

math.AP↗
arXiv2007

A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ $0\le u <\fracπ2$, $0\le v <π$, is the \emph{unique} extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.

math.MG↗
arXiv2007

A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated to this equation. We prove that the solution blows up at finite time $T$ if and only if its energy is negative at some time before $T$. The proof of this result is based on a Gamma-convergence technique.

math.AP↗
arXiv2005

Greatest least eigenvalue of the Laplacian on the Klein bottle

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$λ\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 πE(2\sqrt 2/3),$$ where $λ\_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of the Laplacian and the area of $(\mathbb{K},g)$, respectively, and $E$ is the complete elliptic integral of the second kind. Moreover, the equality is uniquely achieved, up to dilatations, by the metric $$g\_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ with $0\le u,v <π$. The proof of this theorem leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.

math.MG↗