Mathematics

In this paper we investigate the asymptotic behavior and decay of the solution of the discrete in time N -dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the asymptotic decay, both in $L^p(\R^N).$ Furthermore we prove optimal L 2 -decay of solutions. Since the technique of energy methods is not applicable, we follow the approach of estimates based on the discrete fundamental solution which is given by an original integral representation and also by MacDonald's special functions. As a consequence, the analysis is different to the continuous in time heat equation and the calculations are rather involved.

We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Ampère type equations that origin from mean curvature equations of Lagrangian graph (x,Du(x)) and refine the previous study on zero mean curvature equations and the Monge-Ampère equations.

We consider the Hénon equation ?�Δu=|x | α |u | p?? u in B N ,u=0 on ??B N , ( P α ) where B N ??R N is the open unit ball centered at the origin, N?? , p>1 and α>0 is a parameter. We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation ?�Δw=|w | p?? win B 2 ,w=0on ??B 2 , where B 2 ??R 2 is the open unit ball, is the limit problem of ( P α ) , as α?��? , in the framework of radial solutions. We exploit this fact to prove several qualitative results on the radial solutions of ( P α ) with any fixed number of nodal sets: asymptotic estimates on the Morse indices along with their monotonicity with respect to α ; asymptotic convergence of their zeros; blow up of the local extrema and on compact sets of B N . All these results are proved for both positive and nodal solutions.

We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equation and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable x/t and in which the time and space derivatives are coupled together. We will first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable s=x/t . In terms of the standard Fisher-KPP equation, our results leads to a new class of "asymptotically homogeneous" environments which share the same spreading speed with the corresponding homogeneous environments.

We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space H 1 ? L 2 . This model has solitary waves with speeds ??<c<1 . When |c| approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.

In this paper, we consider the exogenous chemotaxis system with physical mixed zero-flux and Dirichlet boundary conditions in one dimension. Since the Dirichlet boundary condition can not contribute necessary estimates for the cross-diffusion structure in the system, the global-in-time existence and asymptotic behavior of solutions remain open up to date. In this paper, we overcome this difficulty by employing the technique of taking anti-derivative so that the Dirichlet boundary condition can be fully used, and show that the system admits global strong solutions which exponentially stabilize to the unique stationary solution as time tends to infinity against some suitable small perturbations. To the best of our knowledge, this is the first result obtained on the global well-posedness and asymptotic behavior of solutions to the exogenous chemotaxis system with physical boundary conditions.

In this paper, we study the super-critical Quasi-Geostrophic equation in Gevrey-Sobolev space. We prove the local existence of (QG) for any large initial data and we give an exponential type of Blow-up to the solution. Moreover, we establish the existence global for a small initial data and we show that ?��???H s a, α ?? decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.

In this paper we prove, if θ?�C([0,??, H 2??α ( R 2 )) is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then ?��?t) ??H 2??α decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.

In this paper, we study various aspects of the ODE's flow X solution to the equation ??t X(t,x)=b(X(t,x)) , X(0,x)=x in the d -dimensional torus Y d , where b is a regular Z d -periodic vector field from R d in R d .We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field b : (i) the everywhere asymptotics of the flow X , (ii) the almost-everywhere asymptotics of the flow X , (iii) the global rectification of the vector field b in Y d , (iv) the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, (v) the unit set condition for Herman's rotation set C b composed of the means of b related to the invariant probability measures, (vi) the unit set condition for the subset D b of C b composed of the means of b related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, (vii) the homogenization of the linear transport equation with oscillating data and the oscillating velocity b(x/ε) when b is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow X and the unit set condition for D b are equivalent when D b is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when b is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any d -dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

We consider the nonlinear damped Klein-Gordon equation ??tt u+2α ??t u?�Δu+u?�|u | p?? u=0on [0,??? R N with α>0 , 2?�N?? and energy subcritical exponents p>2 . We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times. We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate t ?? to the excited state.