Initial value problems for nonlinear heat equations

title: Partial regularity for weak heat flows into spheres
abstract: In this paper we show that a weak heat flow of harmonic maps from a compact Riemannian manifold (possibly with boundary) into a sphere, satisfying the monotonicity inequality and the energy inequality, is regular off a closed set of m-dimensional Hausdorff measure zero.

title: Evolution of harmonic maps with Dirichlet boundary conditions
abstract: In this paper we shall study a left over problem concerning the heat flow of harmonic maps on manifolds with boundary. Let (M, g) be a compact smooth m-dimensional Reimannian manifold with nonempty smooth boundary <9M, and let (iV, h) be a compact smooth n-dimensional Reimannian manifold without boundary. We denote M U dM by M. Since (AT, h) can be isometrically embedded into an Euclidean space M, for some k > n, we may view TV as a submanifold of R. In local coordinates on M, the energy of a map u : M —> N ^-> R^ is given by

title: Evolution equations with a free boundary condition
abstract: In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂1,u −Δu = Γ(u)(∇u, ∇u) [tT TuuN, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥u Tu(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = uo(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M.Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝn, in which the nonlinear equation reduces to ∂tu-Δu = 0.We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than KN <-0 and Σ is totally geodesic in N.

title: Global Classical Solutions for Nonlinear Evolution Equations
abstract: This text represents the results originally obtained by S. Lainerman, D. Christodoulou, Y. Choquet-Bruhat, T. Nishida and A. Matsumara on the global existence of classical solutions to the Cauchy problem with small initial data for nonlinear evolution equations.

title: On existence theorems of periodic traveling wave solution to the generalized forced Kadomtsev-Petviashvili equation
abstract: T uts PAPER is concerned with periodic travcling wave solutions of the generalized forced KadomtsevPetviashvili equation in the form ( u 1 + If( tt)lx + o u x.rx )"' + {3uyy = ho. The basic approach to this problem is to establish an equivalence relat ionship between a periodic boundary value problem and nonlinear integral equations wirh symrnerric kernels by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous periodic functions with a given period 2T. Schauder's fixed point theorem is then used to prove the existence of no nconstant periodic traveling wave solutions.

title: Der Satz von Miquel in der Möbiusebene

title: A nonexistenee theorem of small periodic traveling wave solutions to the generalized Boussinesq equation
abstract: THE GENERALIZED Boussinesq equation, u tt - u xx + [f(u)] xx + u xxxx = 0, and its periodic traveling wave solutions are considered. Using the transform z = x - wt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein-type integral equation is then established by using the Green's function method. This integral equation generates compact operators in (C 2T , ∥.∥), a Banach space of real-valued continuous periodic functions with a given period 2T. We prove that for small T > 0, there exists an r > 0 such that there is no nontrivial solution to the integral equation in the ball B(0,r) ⊆ C 2T . And hence, the generalized Boussinesq equation has no 2T-periodic traveling wave solutions having amplitude less than r.

title: On the Vibration of Beams or Rods Carrying a Concentrated Mass

title: On the inverse scattering problem for the Helmholtz equation in one dimension
abstract: A scheme is presented for the solution of inverse scattering problems for the one-dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the potential, and can be viewed as a frequency-domain version of the layer-stripping approach. The principal advantage of the procedure is that if the scatterer is be reconstructed has m>or=1 continuous derivatives, the accuracy of the reconstruction is proportional to 1/am, where a is the highest frequency for which scattering data are available. Thus a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has an asymptotic cost O(n2), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated by several examples.