We prove convergence of the solutions Xn of semilinear stochastic evolution equations on a Banach space B, driven by a cylindrical Brownian motion in a Hilbert space H, dXn(t) = (AnX(t) + Fn(t,Xn(t))) dt+Gn(t,Xn(t)) dWH(t), Xn(0) = ξn, assuming that the operators An converge to A and the locally Lipschitz functions Fn and Gn converge to the locally Lipschitz functions F and G in an appropriate ...

We will survey some recent results on construction of entire solutions of semilinear elliptic equations. We will mostly focus on the construction of families of solutions to the Allen-Cahn equation of phase transitions, whose level sets suitable scaled concentrate around a given minimal surface. To do so, we shall introduce an infinite-dimensional form of Lyapunov-Schmidt reduction suitable for...

The precise dynamics of a reaction-diffusion model of autocatalytic chemical reaction is described. It is shown that exactly either one, two, or three steady states exists, and the solution of dynamical problem always approaches to one of steady states in the long run. Moreover it is shown that a global codimension one manifold separates the basins of attraction of the two stable steady states....

We study the existence of positive solutions for a fourth order semilinear elliptic equation under Navier boundary conditions with positive, increasing and convex source term. Both bounded and unbounded solutions are considered. When compared with second order equations, several differences and difficulties arise. In order to overcome these difficulties new ideas are needed. But still, in some ...

We consider a pointwise tracking optimal control problem for semilinear elliptic partial differential equation. derive the existence of solutions and analyze first and, necessary sufficient, second order optimality conditions. devise two strategies discretization to approximate solution problem: semidiscrete scheme where variable is not discretized---the so-called variational approach---and ful...