An attractor for the singularly perturbed Kirchhoff equation with supercritical nonlinearity

We consider a quasilinear wave equation of Kirchhoff type

?_utt−(1+‖∇u‖²)Δu+_ut+f(u)=g(x),$?u_{tt}-(1+{\Vert \mathrm{\nabla }u\Vert }^2)\mathrm{\Delta }u+u_t+f(u)=g(x),$

where ?>0$?>0$ is a small parameter. Without any growth restrictions on the nonlinearity f(u)$f(u)$, we prove the existence of a finite-dimensional global attractor on an appropriate (bounded) phase space. The key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation.     

Liouville type results and a maximum principle for non-linear differential operators on the Heisenberg group

We prove Liouville type results for non-negative solutions of the differential inequality _Δφu?f(u)?(|_∇0u|)${\mathrm{\Delta }}_{\phi }u?f(u)?(|{\mathrm{\nabla }}_{0}u|)$ on the Heisenberg group under a generalized Keller–Osserman condition. The operator _Δφu${\mathrm{\Delta }}_{\phi }u$ is the φ  -Laplacian defined by _div0(|_∇0u|⁻¹φ(|_∇0u|)_∇0u)${\mathrm{div}}_0({|{\mathrm{\nabla }}_{0}u|}^{-1}\phi (|{\mathrm{\nabla }}_{0}u|){\mathrm{\nabla }}_{0}u)$ and φ, f and ? satisfy mild structural conditions. In particular, ? is allowed to vanish at the origin. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential inequality.     

Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type

We establish symmetrization results for the solutions of the linear fractional diffusion equation _∂tu+(−Δ)^σ/2u=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}u=f$ and its elliptic counterpart hv+(−Δ)^σ/2v=f$hv+{(-\mathrm{\Delta })}^{\sigma /2}v=f$, h>0$h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, _∂tu+(−Δ)^σ/2A(u)=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}A(u)=f$, but only when the nondecreasing function A:_R+→_R+$A:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(−Δ)^σ/2v=f$B(v)+{(-\mathrm{\Delta })}^{\sigma /2}v=f$ when B(v)$B(v)$ is a convex nonnegative function for v>0$v>0$ with B(0)=0$B(0)=0$, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B   is concave. Such counterexamples do not exist in the standard diffusion case σ=2$\sigma =2$.     

Traveling wave solutions of the heat flow of director fields

We consider the simplest possible heat equation for director fields, ut=Δu+|∇u|²u$u_t=\mathrm{\Delta }u+{|\mathrm{\nabla }u|}^{2}u$ (|u|=1$|u|=1$), and construct axially symmetric traveling wave solutions defined in an infinitely long cylinder. The traveling waves have a point singularity of topological degree 0 or 1.     

Global unique continuation from a half space for the Schrödinger equation

We obtain a global unique continuation result for the differential inequality |(i_∂t+Δ)u|?|V(x)u|$|(i{\partial }_t+\mathrm{\Delta })u|?|V(x)u|$ in Rⁿ⁺¹${\mathbb{R}}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials V(x)$V(x)$ in Rⁿ${\mathbb{R}}^n$. Our method is based on a new type of Carleman estimates for the operator i_∂t+Δ$i{\partial }_t+\mathrm{\Delta }$ on Rⁿ⁺¹${\mathbb{R}}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.     

On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain

This paper is devoted to the Cauchy problem for the nonlinear Schrödinger equation with time-dependent loss/gain which reads i_ut+Δu+λ|u|^αu+ia(t)u=0$i{u}_t+\mathrm{\Delta }u+\lambda {|u|}^{\alpha }u+ia(t)u=0$. This equation appears in the recent studies of Bose–Einstein condensates and optical systems. We obtain some global existence and blow-up results which depend on the size of the loss/gain coefficient. In particular, we prove the global existence for the energy critical nonlinearity. By scaling and compactness arguments, we also discuss asymptotic profiles and concentration properties of blow-up solutions.     

Boundedness of global solutions of a porous medium equation with a localized source

In this paper a localized porous medium equation _ut=u^r(Δu+af(u(_x0,t)))$u_t=u^r(\mathrm{\Delta }u+\mathit{af}(u(x_0,t)))$ is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.     

Morse theory for indefinite nonlinear elliptic problems

Using the heat flow as a deformation, a Morse theory for the solutions of the nonlinear elliptic equation:

−Δu−λu=a₊(x)|u|^q−1u−a₋(x)|u|^p−1u+h(x,u)$-\mathrm{\Delta }u-\lambda u=a_{+}(x){|u|}^{q-1}u-a_{-}(x){|u|}^{p-1}u+h(x,u)$

Higher order energy decay for damped wave equations with variable coefficients

Under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c(x)_utt−div(A(x)∇u)+a(x)_ut=0$c(x)u_{tt}-\mathrm{div}(A(x)\mathrm{\nabla }u)+a(x)u_t=0$ in Rⁿ${\mathbb{R}}^n$ are established. The results concern weighted (in time) and pointwise (in time) energy decay estimates. We also obtain weighted L²$L^2$ estimates for spatial derivatives.