Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity

We consider the blowup rate of solutions for a semilinear heat equation

     

A sufficient condition for type I blowup in a parabolic-elliptic system

This paper is concerned with blowup of positive solutions to a Cauchy problem for a parabolic-elliptic system

     

“Bubble-tower” radial solutions in the slightly supercritical Brezis-Nirenberg problem

In this paper, we consider the Brezis-Nirenberg problem in dimension N?4, in the supercritical case. We prove that if the exponent gets close to and if, simultaneously, the bifurcation parameter tends to zero at the appropriate rate, then there are radial solutions which behave like a superposition of bubbles, namely solutions of the form

     

Uniqueness of positive radial solutions for Δu−u+u=0 on an annulus

We prove uniqueness of positive radial solutions to the semilinear elliptic equation , subject to the Dirichlet boundary condition on an annulus in . As a by-product, our argument also provides a much simpler, if not the simplest, new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space .     

Initial blow-up rates and universal bounds for nonlinear heat equations

We establish a universal upper bound on the initial blow-up rate for all positive classical solutions of the Dirichlet problem for the nonlinear heat equation

     

Pressure integrability conditions for uniqueness of mild solutions of the Navier-Stokes system

We address uniqueness of mild solutions of the Navier-Stokes system in . We prove that solutions for which the pressure is locally square integrable are unique.     

Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation on D×(0,∞), where is a some unbounded domain in and V belongs to a new parabolic class J^∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P^∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal.     

Stability of perturbed nonlinear parabolic equations with Sturmian property

We study stability of an equilibrium f∗ of autonomous dynamical systems under asymptotically small perturbations of the equation. We show that such stability takes place if the domain of attraction of the equilibrium f∗ contains a one-parametric ordered family . In the stability analysis we need a special S-relation (a kind of “restricted partial ordering”) to be preserved relative to the family . This S-relation is inherited from the Sturmian zero set properties for linear parabolic equations. As main applications, we prove stability of the self-similar blow-up behaviour for the porous medium equation, the p-Laplacian equation and the dual porous medium equation in with nonlinear lower-order perturbations. For such one-dimensional parabolic equations the S-relation is Sturm's Theorem on the nonincrease of the number of intersections between the solutions and particular solutions with initial data in . This Sturmian property plays a key role and is true for the unperturbed PME, but is not true for perturbed equations.     

Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents

We describe special asymptotic structures of solutions of the semilinear heat equation

     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.     

Nonlinear elliptic equations with subhomogeneous potentials

We prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation

     

On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation

We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation

     

Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type

We find for small ε positive solutions to the equation

     

Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations

Sufficient and necessary conditions for the embeddings between Besov spaces and modulation spaces are obtained. Moreover, using the frequency-uniform decomposition method, we study the Cauchy problem for the generalized BO, KdV and NLS equations, for which the global well-posedness of solutions with the small rough data in certain modulation spaces is shown.     

Multibump nodal solutions for an indefinite superlinear elliptic problem

We define some Nehari-type constraints using an orthogonal decomposition of the Sobolev space and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem.     

Radially symmetric weak solutions for a quasilinear wave equation in two space dimensions

We prove the convergence of the radially symmetric solutions to the Cauchy problem for the viscoelasticity equations

     

Constructing topologically distinct energy-critical curves in the path space of the Euclidean line

We construct topologically distinct global, non-embedding solutions to the Euler-Lagrange equation for a natural energy functional on the space of maps .     

Large-time geometric properties of solutions of the evolution p-Laplacian equation

We establish the behavior of the solutions of the degenerate parabolic equation