The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H1 (L2 norm and energy). We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the...

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (i...

Lagrangian spheres in the symplectic Del Pezzo surfaces arising as blow-ups of CP in 4 or fewer points are classified up to Lagrangian isotopy. Unlike the case of the 5-point blow-up, there is no Lagrangian knotting.

In this work, we study the blow-up and global solutions for a quasilinear reaction–diffusion equation with a gradient term and nonlinear boundary condition:      (g(u)) t = ∆u + f (x, u, |∇u| 2 , t) in D × (0, T), ∂u ∂n = r(u) on ∂D × (0, T), u(x, 0) = u 0 (x) > 0 in D, where D ⊂ R N is a bounded domain with smooth boundary ∂D. Through constructing suitable auxiliary functions and using ma...

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 show that nonlinear Liouville theorems does not hold in general for indefinite problems on half spaces. Thus, in order to use blow-up method to obtain a priori estimates of indefinite elliptic equations, one has to impose assumptions on the nodal set of nonlinearity. The counter example is constructed by shooting method in one-dimensional case and then extended to higher dimensions.

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.