We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in L p for p > 4 3 . We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on R 4 . As a corollary, we obtain an energy identity for the heat flow of biharmonic maps at time infinity.

Abstract. Let Iν (x) be the modified Bessel functions of the first kind of order ν, and Sp,ν (x) = Wν (x) 2

In this paper, we restrict our attention to traveling wave solutions of a reactiondiffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.

Abstract. Let I v (x) be modified Bessel functions of the first kind. We prove the monotonicity property of the function x →