Juan J. L. Velázquez

Singularity patterns in a chemotaxis model

The authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time. Such a finite-time blow-up of solutions implies chemotactic collapse, namely, concentration of species to form sporae. The model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity, which has the form ut=Δu−χ(uv), 0=Δv+(u−1), on ΩR2, where Ω is an open disc with no-flux (homogeneous Neumann) boundary conditions. The initial conditions are continuous functions u(x,0)=u0(x)≥0, v(x,0)=v0(x)≥0 for xΩ. Under these conditions, the authors prove there exists a radially symmetric solution u(r,t) which blows up at r=0, t=T<∞. A specific description of such a solution is presented. The authors also discuss the strong similarity between the chemotactic model they study and the classical Stefan problem.

Blow-up behaviour of one-dimensional semilinear parabolic equations

Abstract Consider the Cauchy problem u t − u x x − F ( u ) = 0 ; x ∈ ℝ , t > 0 u ( x , 0 ) = u 0 ( x ) ; x ∈ ℝ where u0(x) is continuous, nonnegative and bounded, and F(u) = up with p > 1, or F(u) = eu. Assume that u blows up at x = 0 and t = T > 0. In this paper we shall describe the various possible asymptotic behaviours of u(x, t) as (x, t) → (0, T). Moreover, we shall show that if u0(x) has a single maximum at x = 0 and is symmetric, u0(x) = u0(−x) for x > 0, there holds 1) If F(u) = up with p > 1, then lim t ↑ T u ( ξ ( ( T − t ) | log ( T − t ) | ) 1 / 2 , t ) × ( T − t ) 1 / ( p − 1 ) = ( p − 1 ) − ( 1 / ( p − 1 ) ) [ 1 + ( p − 1 ) ξ 2 4 p ] − ( 1 / ( p − 1 ) ) uniformly on compact sets |ξ| ≦ R with R > 0, 2) If F(u) = eu, then lim t ↑ T ( u ( ξ ( ( T − t ) | log ( T − t ) | ) 1 / 2 , t ) + log ( T − t ) ) = − log [ 1 + ξ 2 4 ] uniformly on compact sets |ξ| ≦ R with R > 0.

Finite-time aggregation into a single point in a reaction - diffusion system

We consider the following system: which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dirac mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed.

Self-similar blow-up for a reaction-diffusion system

Abstract This work is concerned with the following system: which is a model to describe several phenomena in which aggregation plays a crucial role as, for instance, motion of bacteria by chemotaxis and equilibrium of self-attracting clusters. When the space dimension N is equal to three, we show here that (S) has radial solutions with finite mass that blow-up in finite time in a self-similar manner. When N = 2, however, no radial solution with finite mass may give rise to self-similar blow-up.

Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

This paper is concerned with positive solutions of the semilinear system: ut=δu+vp, p⩾1,vt=δv+uq, q⩾1,(S) which blow up at x = 0 and t = T < ∞. We shall obtain here conditions on p, q and the space dimension N which yield the following bounds on the blow up rates: u(x,t) ⩽ C(T−t)−p+1pq−1, v(x,t)⩽C(T−t)−q+1pq−1, (1) for some constant C > 0. We then use (1) to derive a complete classification of blow up patterns. This last result is achieved by means of a parabolic Liouville theorem which we retain to be of some independent interest. Finally, we prove the existence of solutions of (S) exhibiting a type of asymptotics near blow up which is qualitatively different from those that hold for the scalar case.

The Becker–Döring Equations and the Lifshitz–Slyozov Theory of Coarsening

In this paper the relation between the kinetic set of Becker–Döring (BD) equations and the classical Lifshitz–Slyozov (LS) theory of coarsening is studied. A model that resembles the LS theory but keeps some of the nucleation effects is derived. For this model a solution is described that shows how the kinetic effects explain the particular solution selected in the LS theory. By means of a renormalization procedure, a discrete group of transformations is shown to play an important role in describing the structure of the solution near the critical size of the LS theory.

Blow–Up Profiles in One–Dimensional. Semilinear Parabolic Problems

Partially supporte by CICYT Research Grant PB86–0112–00202 and EEC Contrast SCI–0019–c. We consider the Cauchy Problem where p > 1 and U

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

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On the melting of ice balls

esub:(x) is continuous, nonnegative and bounded. Let u(x.t) be the solution of (1). (2). and assume that u blows up at t=T . and We then show that the blow–up set is discrete. Moreover, if x=0 is a blow–up point, one of the two following possibilities occurs. Either There exist c > 0 and an even number m such that

Approaching an extinction point in one-dimensional semilinear heat-equations with strong absorption

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.