M. Escobedo

Boundedness and blow up for a semilinear reaction-diffusion system

Abstract We consider the semilinear parabolic system (S) u t − Δu = ν p ν t − Δν = u q , where x∈RNN ⩾ 1, t > 0, and p, q are positive real numbers. At t = 0, nonnegative, continuous, and bounded initial values (u0, v0(x)) are prescribed. The corresponding Cauchy problem then has a nonnegative classical and bounded solution (u(t, x), v(t,x)) in some strip ST = [0,T) × R N, 0 T∗ = sup {T > 0:u, v remain bounded in S T } . We show in this paper that if 0 T∗ = + ∞ , so that solutions can be continued for all positive times. When pq > 1 and (γ + 1) (pq − 1) ⩾ N 2 with γ = max {p, q}, one has T∗ for every nontrivial solution (u, v). T∗ is then called the blow up time of the solution under consideration. Finally, if (γ + 1)(pq − 1) N 2 both situations coexist, since some nontrivial solutions remain bounded in any strip SΓ while others exhibit finite blow up times.

A semilinear parabolic system in a bounded domain

SummaryConsider the system(S)n

A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma

left{ begin{gathered} u_t - Delta u = v^p , in Q = { (t, x), t > 0, x in Omega } , hfill v_t - Delta v = u^q , in Q , hfill u(0, x) = u_0 (x) v(0, x) = v_0 (x) in Omega , hfill u(t, x) = v(t, x) = 0 , when t geqslant 0, x in partial Omega , hfill end{gathered} right.

Radiation dynamics in homogeneous plasma

n where Ω is a bounded open domain in ℝN with smooth boundary, p and q are positive parameters, and functions u0 (x), v0(x) are continuous, nonnegative and bounded. It is easy to show that (S) has a nonnegative classical solution defined in some cylinder QT=(0, T) × Ω with T ⩽ ∞. We prove here that solutions are actually unique if pq ⩾ 1, or if one of the initial functions u0, v0 is different from zero when 0 < pq < 1. In this last case, we characterize the whole set of solutions emanating from the initial value (u0, v0)=(0, 0). Every solution exists for all times if 0 1, solutions may be global or blow up in finite time, according to the size of the initial value (u0, v0).

On self-similarity and stationary problem for fragmentation and coagulation models

Abstract The occurrence of gelation and the existence of mass-conserving solutions to the continuous coagulation–fragmentation equation are investigated under various assumptions on the coagulation and fragmentation rates, thereby completing the already known results. A non-uniqueness result is also established and a connection to the modified coagulation model of Flory is made.

Gelation in Coagulation and Fragmentation Models

This work deals with the problem consisting in the equation (1) partial derivative f/partial derivative t = 1/x(2) partial derivative/partial derivative x [x(4)(partial derivative f/partial derivative x + f + f(2))], when x is an element of (0, infinity), t > 0, together with no-flux conditions at x = 0 and x = +infinity, i.e. (2) x(4)( partial derivative f/partial derivative x + f + f(2))=0 as x --> 0 or x --> +infinity. Such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution f(x,t) in a homogeneous plasma when radiation interacts with matter via Compton scattering. We shall prove that there exist solutions of (1), (2) which develop singularities near x = 0 in a finite time, regardless of how small the initial number of photons N(0) = integral(0)(+infinity) x(2) f(x, 0)dx is. The nature of such singularities is then analyzed in detail. In particular, we show that the flux condition (2) is lost at x = 0 when the singularity unfolds. The corresponding blow-up pattern is shown to be asymptotically of a shock wave type. In rescaled variables, it consists in an imploding travelling wave solution of the Burgers equation near x = 0, that matches a suitable diffusive profile away from the shock. Finally, we also show that, on replacing (2) near x = 0 as determined by the manner of blow-up, such solutions can be continued for all times after the onset of the singularity.

Homogeneous Boltzmann equation in quantum relativistic kinetic theory

Let (u(t, x), v(t, x)) and (uBAR(t, x), vBAR(t, x)) be two nonnegative classical solutions of (S)[GRAPHICS:{ut=Δu+vp, p>0 ; vt=Δv+uq, q>0] in some strip S(T) = (0, T) x R(N), where 0 < T ≤ ∞, and suppose that u(0, x) = uBAR(0, x), v(0, x) = vBAR(0, x), where u(0, x) and v(0, x) are continuous, nonnegative, and bounded real functions, one of which is not identically zero. Then one has u(t, x) = uBAR(t, x), v(t, x) = vBAR(t, x) in S(T). If pq ≥ 1, the result is also true if u(0, x) = v(0, x) = 0. On the other hand, when 0 < pq < 1, the set of solutions of (S) with zero initial values is given by u(t; s) = c1(t - s)+(p+1)/(1-pq), v(t; s) = c2(t - s)+(q+1)/(1-qp), where 0 ≤ s ≤ t, c1 and c2 are two positive constants depending only on p and q, and (ξ)+ = max{ξ,0}.

Asymptotic description of Dirac mass formation in kinetic equations for quantum particles

Abstract We study in this paper the asymptotic behaviour of solutions of a nonlinear Fokker–Plank equation. Such an equation describes the evolution of radiation for a gas of photons, which interacts with electrons by means of Compton scattering and Bremsstrahlung radiation. Assuming that a suitable adimensional parameter e (which measures the strength of the Bremsstrahlung effect) is small enough, we show that the problem considered has two natural timescales. For times t=O(1), the dynamics is conducted by that of a reduced problem, corresponding to setting e=0 in the original equations. Solutions of that problem may blow up in finite time, and the total number of photons is no longer preserved after the singularity formation. Nevertheless, solutions of this problem can be continued for all times, if defined in a suitable sense. When t→∞, solutions of such a modified problem converge towards a Bose–Einstein distribution with a suitable (in general nonzero) chemical potential. However, at times of order t= O ((e∣ log e∣) −2/3 ) , the Bremsstrahlung term becomes dominant at low frequencies, and drives the photon distribution to approach to a Planck distribution as time goes to infinity.