Volume 37, Issue 2

In this paper we study the semilinear elliptic problem -Δ⁢u-k2⁢u=Q⁢|u|p-2⁢u in ⁢ℝ2,-\Delta u-k^{2}u=Q|u|^{p-2}u\quad\text{in }\mathbb{R}^{2}, where k>0{k>0}, p≥6{p\geq 6} and Q is a bounded function. We prove the existence of real-valued W2,p{W^{2,p}}-solutions, both for decaying and for periodic coefficient Q . In addition, a nonlinear far-field relation is derived for these solutions.

We consider a Dirac-type equation with a quadratic nonlinearity with initial data in the Gevrey spaces Gσ,s{G^{\sigma,s}}, whose functions admit analytic continuations. We show that the Cauchy problem is locally well-posed in Gσ,s{G^{\sigma,s}}, thus giving us short-time persistence of analyticity of solutions. In addition, we prove sufficient conditions for the solutions to be analytic for all time.

We consider a singularly perturbed fourth-order problem with third-order terms on the unit square. With a formal power series approach, we decompose the solution into solutions of reduced (third-order) problems and various layer parts. The existence of unique solutions for the problem itself and for the reduced third-order problems is also addressed. To our knowledge, this paper is a first attempt for a solution decomposition of such problems.

We consider a model for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The model consists of an ordinary differential equation for the prion monomers and a hyperbolic nonlinear differential equation with integral terms for the prion polymers and was shown to possess global weak solutions for unbounded reaction rates [11]. Here we prove the uniqueness of weak solutions.