We analyze dissipative boundary conditions for nonlinear hyperbolic systems in one space dimension. We show that a known sufficient condition for exponential stability with respect to the H2-norm is not sufficient for the exponential stability with respect to the C1-norm. Hence, due to the nonlinearity, even in the case of classical solutions, the exponential stability depends strongly on the n...

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in [Li], we partially extend known results for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list the ones we feel particularly interesting in the final section. Mathematics Subject Classification: 35J65; 35J40.

Consider the problem  ∆u = λe in B u = ∂u ∂n = 0 on ∂B, where B is the unit ball in R and λ is a parameter. Unlike the Gelfand problem the natural candidate u = −4 ln(|x|), for the extremal solution, does not satisfy the boundary conditions and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Dávila et al. in [5] use...

where Ω is a bounded, smooth domain in R, φ1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h ∈ C(Ω̄). We prove that given k ≥ 1 this problem has at least k solutions for all sufficiently large s > 0, which answers affirmatively a conjecture by Lazer and McKenna [22] for this case. The solutions found exhibit multiple concentration behavior around maxi...

We consider classical and entanglement-assisted versions of a distributed computation scheme that computes nonlinear Boolean functions of a set of input bits supplied by separated parties. Communication between the parties is restricted to take place through a specific apparatus which enforces the constraints that all nonlinear, nonlocal classical logic is performed by a single receiver, and th...

We analyze the possible concentration behavior of heat flows related to the Moser-Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser-Trudinger energy in a supercritical regime.

Let λ > 0 denote the largest possible value of λ such that