Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem

We investigate singular and degenerate behavior of solutions of the unstable free boundary problem

Δu=−χ{u>0}.     

Existence and regularity to an elliptic equation with logarithmic nonlinearity

We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in Ω⊂Rn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ^∗ such that uλ vanishes on a set of positive measure for 0<λ<λ^∗, and uλ>0 for λ>λ^∗. If f is concave, for λ>λ^∗ we characterize uλ by its stability.     

Multiplicity of solutions for the plasma problem in two dimensions

Let Ω be a bounded domain in R², u₊=u if u?0, u₊=0 if u<0, u₋=u₊−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma

     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

An inverse problem of identifying the coefficient in a nonlinear parabolic equation

This paper deals with the determination of a pair (p,u) in the nonlinear parabolic equation

u_t−u_xx+p(x)f(u)=0,     

Quasi-periodic solutions in a nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation

iut−uxx+mu+⁴|u|u=0     

Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

Let {A₁,…,AK}⊂Cd×d be arbitrary K matrices, where K and d both ?2. For any 0<Δ<∞, we denote by the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R₊ satisfying tj−tj−1?Δ and

     

A Liouville type theorem for a variational problem with free boundary in three dimensions

We consider the minimum problem for the functional

_EΩ(u)=_∫Ω(|Du|²+λ²_χ{u>0})$E_{\Omega }\left(u\right)={\int }_{\Omega }({\left|Du\right|}^2+{\lambda }^2{\chi }_{\{u>0\}})$

in three dimensional space, where λ>0$\lambda >0$ is a constant. We will establish a Liouville type theorem for this variational problem: if u∈C(R³)$u\in C\left({\mathbb{R}}^3\right)$ is a nonnegative and nonzero global minimizer, then u(x)=λ((x−_x0)⋅ν)⁺$u\left(x\right)=\lambda {((x-x_0)\cdot \nu )}^{+}$ for some point _x0$x_0$ and some unit vector ν$\nu$.     

Deduction of L. Hörmander's extension of Ásgeirsson's mean value theorem

L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δ_x−Δ_y)u=f, , , then

     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

On the solutions of the Cauchy problem for a class of partial differential equations with double characteristic at a point

We give an explicit representation of the solutions of the Cauchy problem, in terms of series of hypergeometric functions, for the following class of partial differential equations with double characteristic at the origin:

(xkt_∂+ax_∂)(xkt_∂+bx_∂)u+cxk−1t_∂u=0,     

The Brezis-Nirenberg problem near criticality in dimension 3

We consider the problem of finding positive solutions of Δu+λu+u^q=0 in a bounded, smooth domain Ω in , under zero Dirichlet boundary conditions. Here q is a number close to the critical exponent 5 and 0<λ<λ₁. We analyze the role of Green's function of Δ+λ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either q or λ are regarded as parameters. As a special case of our results, we find that if , where λ∗ is the Brezis-Nirenberg number, i.e., the smallest value of λ for which least energy solutions for q=5 exist, then this problem is solvable if q>5 and q−5 is sufficiently small.     

Lipschitz metric for the periodic Camassa-Holm equation

We study the stability of conservative solutions of the Cauchy problem for the Camassa-Holm equation ut−uxxt+κux+3uux−2uxuxx−uuxxx=0 with periodic initial data u₀. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t),v(t))?eCtdD(u₀,v₀). The relationship between this metric and usual norms in and is clarified.     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

Mathematical analysis to a nonlinear fourth-order partial differential equation

The paper first study the steady-state thin film type equation

∇⋅(uⁿ|∇Δu|^q−2∇Δu)−δu^mΔu=f(x,u)     

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$

on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u₀(x) = u(x, 0) and the boundary data g₀(t) = u(0, t), g₁(t) = u_x(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

     

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation of the form ut=Δu+ε⁻²fε(x,t,u), where ε is a small parameter and fε(x,t,u)=f(u)−εgε(x,t,u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u₀ that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε²|lnε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where gε≡0.Next we consider systems of reaction-diffusion equations of the form

     

Global bifurcation for nonlinear equations

Using a bifurcation result on noncompact branches of solutions in an abstract setting, we establish the existence of global bifurcation for the following nonlinear equation

−div(a|∇u|^p−2∇u)−_μ0b|u|^p−2u=q(λ,x,u,∇u)$-\text{div}\left(a{\left|\nabla u\right|}^{p-2}\nabla u\right)-{\mu }_{0}b{\left|u\right|}^{p-2}u=q(\lambda ,x,u,\nabla u)$