Miguel Escobedo

Large time behavior for convection-diffusion equations in RN

Abstract We describe the large time behavior of solutions of the convection-diffusion equation u t − Δu = a · ▽(¦u¦ q − 1 u) in (0, ∞) × R N with a ϵ R N and q ⩾ 1 + 1 N , N ⩾ 1 . When q = 1 + 1 N , we prove that the large time behavior of solutions with initial data in L 1 ( R N ) is given by a uniparametric family of self-similar solutions. The relevant parameter is the mass of the solution that is conserved for all t . Our result extends to dimensions N > 1 well known results on the large time behavior of solutions for viscous Burgers equations in one space dimension. The proof is based on La Salles Invariance Principle applied to the equation written in its self-similarity variables. When q > 1 + 1 N the convection term is too weak and the large time behavior is given by the heat kernel. In this case, the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem.

Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations

AbstractLet D ⊂ RN be either all of Rn or else a cone in RN whose vertex we may take to be at the origin, without loss of generality. Let pi, qj, i = 1, 2, be nonnegative with 0<p1+q1≦p2+q2. We consider the long-time behavior of nonnegative solutions of the system (S)

On a quantum Boltzmann equation for a gas of photons

u_t = \Delta u + u^{p_1 } v^{q_1 } , v_t = \Delta v + u^{p_2 } v^{q_2 }

Singular solutions for the Uehling-Uhlenbeck equation

in D × [0, ∞) with u0 = v0 = 0 on ∂D, (u, v)t(x,0) = (ν0, ν0)t(x), u0, ν0≧0, u0, ν0 ε L∞(D). We obtain Fujita-type global existence-global nonexistence theorems for (S) analogous to the classical result of Fujita and others for the initial-value problem for ut = Δu + up, u(x, 0) = u0(x) ≧ 0. The principal result in the case D = RN and P2q1 > 0 is that when p1 ≧ 1, the system behaves like the single equation ut=Δu+up1vq1 with respect to Fujita-type blowup theorems, whereas if p1 < 1, the behavior of the system is more complicated. Some of the results extend those of Escobedo & Herrero when D = RN and of Levine when D is a cone. These authors considered (S) in the case of p1 = q2 = 0. An example of nonuniqueness is also given.

Improved intermediate asymptotics for the heat equation

In this paper we investigate the large time behavior of solutions of the semilinear heat equation. Where u{sub 0} is the initial data, N {ge} 1 and p > 1. It can be easily checked that if u(t,x) satisfied (1.1), then for {gamma} > 0 the rescaled functions u{sub {gamma}}(t,x) satisfies (1.1), then for {gamma}>0 the rescaled functions define a one parameter family of solutions to (1.1). A solution u {equivalent_to} 0 is said to be self-similar, when u{sub {gamma}} {equivalent_to} u for all {gamma} > 0. For instance, for any fixed p > 1, w{sup *}(t,x):=((p-1)t){sup {minus}1/(p-1)} is such a solution. Actually, it has been proved by H.Brezis, L.A. Peletier & D. Terman that for 1 {infinity}. 15 refs.

Analytical approach to relaxation dynamics of condensed Bose gases

Abstract We prove existence and uniqueness of the solution of a homogeneous quantum Boltzmann equation describing the photon–electron interaction. We study the asymptotic behaviour of the solutions, and show in particular, that the photon density distribution condensates at the origin asymptotically in time when the total number of photons is larger than a given positive constant. We also recover the Kompaneets equation as a Fokker–Planck type limit of this Boltzmann model.

Fast Reaction Limit of the Discrete Diffusive Coagulation–Fragmentation Equation

Abstract The Keller–Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less 8 π , a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables.

Instability of the Rayleigh-Jeans spectrum in weak wave turbulence theory

In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like k −7/6 as k → 0. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function k −7/6 that we obtained in an earlier paper.