We obtain some conditions under which the positive solution for semidiscretizations of the semilinear equation ut uxx − a x, t f u , 0 < x < 1, t ∈ 0, T , with boundary conditions ux 0, t 0, ux 1, t b t g u 1, t , blows up in a finite time and estimate its semidiscrete blow-up time. We also establish the convergence of the semidiscrete blow-up time and obtain some results about numerical blow-u...

After a brief discussion of known global well-posedness results for semilinear systems, we introduce a class of quasilinear systems and obtain spatially local estimates which allow us to prove that if one component of the system blows up in finite time at a point x∗ in space then at least one other component must also blow up at the same point. For a broad class of systems modelling one-step re...

We consider the L 2-gradient ow associated with the Yang-Mills functional, the so-called Yang-Mills heat ow. In the setting of a trivial principal SO(n)-bundle over R n in dimension n greater than 4, we show blow-up in nite time for a class of SO(n)-equivariant initial connections.

The singularly perturbed boundary blow-up problem −ε2∆u = u(u− a)(1− u) u > 0 in B, u = ∞ on ∂B is studied in the unit ball B ⊂ R (N ≥ 2), a ∈ (1/2, 1) is a constant. It is shown that there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions.

By blow{up analysis on the descending ow for the Yamabe quotient of a compact Reimannian manifold (M n ; g); n 3 , we prove that the ow globally exists and subconverges to u1 such that the scalar curvature of u 4 n?2

For certain singularly perturbed two-component reaction-diffusion (RD) systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behavior in terms of some parameter β in the system. For some such systems, such as the Gray-Scott model, a spike self-replication behavior is observed as a parameter varies across the saddle-node point. We demonstrate and anal...

We investigate in this article the long-time behaviour of the solutions to the energydependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under statistical description, that dissipates energy during collisions. We assume that the gas is “anomalous”, in the sense that energy dissipation increases when tempera...

MARÍA JESÚS ÁLVAREZ Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears; Crtra. Valldemossa, km 7.5, 07122, Palma, Spain E-mail: [email protected] ANTONI FERRAGUT Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya; Av. Diagonal, 647, 08028, Barcelona, Catalunya, Spain E-mail: [email protected] XAVIER JARQUE Departament d’Enginyeria Informà...

We study the behaviour of nonnegative solutions of the reaction-diffusion equation    ut = (u)xx + a(x)up in R× (0, T ), u(x, 0) = u0(x) in R. The model contains a porous medium diffusion term with exponent m > 1, and a localized reaction a(x)up where p > 0 and a(x) ≥ 0 is a compactly supported function. We investigate the existence and behaviour of the solutions of this problem in dependenc...

We consider the two dimensional L critical nonlinear Schrödinger equation i∂tu+ ∆u+ u|u| = 0. In the pioneering work [2], Bourgain and Wang have constructed smooth solutions which blow up in finite time T < +∞ with the pseudo conformal speed ‖∇u(t)‖L2 ∼ 1 T − t , and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynam...