Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

The graph grabbing game on Km,n-trees

The graph grabbing game is a two-player game on weighted connected graphs where all weights are non-negative. Two players, Alice and Bob, alternately remove a non-cut vertex from the graph (i.e., the resulting graph is still connected) and get the weight assigned to the vertex, where the starting player is Alice. Each player’s aim is to maximize his/her outcome when all vertices have been taken, and Alice wins the game if she gathered at least half of the total weight. Seacrest and Seacrest (2017) proved that Alice has a winning strategy for every weighted tree with even order, and conjectured that the same statement holds for every weighted connected bipartite graph with even order. In this paper, we prove that Alice wins the game on a type of a connected bipartite graph with even order called a $K_{m,n}$-tree.     

Sur un problème de type elliptique parabolique non linéaireOn a nonlinear elliptic-parabolic problem

We study the general equation b(u)_t=a(u,?(u)_x)_x+f of elliptic-parabolic type. Using the theory of evolution equation governed by accretive operator, we establish existence and uniqueness of mild solutions to the associate Cauchy problem, under general assumptions on the data. With additional structural condition of Alt–Luckhauss type, we show that the mild solutions are weak solutions. To cite this article: S. Ouaro, H. Touré, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 27–30     

On Shimura subvarieties generated by families of abelian covers of P1

We investigate the occurrence of Shimura (special) subvarieties in the locus of Jacobians of abelian Galois covers of ${\mathbb{P}}^1$ in $A_g$ and give classifications of families of such covers that give rise to Shimura subvarieties in the Torelli locus $T_g$ inside $A_g$. Our methods are based on Moonen–Oort works as well as characteristic p techniques of Dwork and Ogus and Monodromy computations.     

A degree condition for a graph to have (a,b)-parity factors

Let $a$ and $b$ be two positive integers such that $a\le b$ and $a\equiv b(mod2)$. A graph $F$ is an $(a,b)$-parity factor of a graph $G$ if $F$ is a spanning subgraph of $G$ and for all vertices $v\in V\left(F\right)$, $d_F\left(v\right)\equiv b(mod2)$ and $a\le d_F\left(v\right)\le b$. In this paper we prove that every connected graph $G$ with $n\ge b(a+b)(a+b+2)∕\left(2a\right)$ vertices has an $(a,b)$-parity factor if $na$ is even, $\delta \left(G\right)\ge (b?a)∕a+a$, and for any two nonadjacent vertices $u,v\in V\left(G\right)$, $max\{d_G\left(u\right),d_G\left(v\right)\}\ge \frac{an}{a+b}$. This extends an earlier result of Nishimura (1992) and strengthens a result of Cai and Li (1998).     

Proof of the Hénon–Lane–Emden conjecture in R3

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on $S^{N?1}$, we prove this conjecture is true for space dimension $N=3$; which also implies the single elliptic equation has no positive classical solutions in ${\mathbb{R}}^3$ when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for ${\mathbb{R}}^3$.     

Multiplicity of solutions for the plasma problem in two dimensions

Let Ω be a bounded domain in R², u₊=u if u?0, u₊=0 if u<0, u₋=u₊−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma

     

Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index

In this paper we study a quasilinear elliptic problem with the Dirichlet boundary condition in a bounded domain involving the operator Au=−Δ_pu−Δ_qu. Assuming that the nonlinearity has a concave-convex behavior we obtain some multiplicity results. More precisely, we obtain five nontrivial solutions; two by a minimization argument, two by the mountain pass theorem, and the other by a cohomological linking theorem.