Existence,Non-existence,and Uniqueness for a Heat Equation with Exponential Nonlinearity in ℝ2

We consider a semilinear heat equation with exponential nonlinearity in ?². We prove that local solutions do not exist for certain data in the Orlicz space exp L ²(?²), even though a small data global existence result holds in the same space exp L ²(?²). Moreover, some suitable subclass of exp L ²(?²) for local existence and uniqueness is proposed.

     

The annealing behavior of hydrogen implanted into Al-1.5 at.% Si alloy

We have studied effects of added elements as well as defects on trap-sites of hydrogen in metals. For the purpose, we observed depth profiles and thermal behaviors of hydrogen implanted into Al-1.5 at.% Si alloy samples in an implantation-temperature range of liquid nitrogen temperature (LNT) to 373 K at different doses. The results were compared with those for pure aluminum samples. It was found that hydrogen is trapped as molecules in grain boundaries of Al/Si.     

Moduli Space of BPS Walls in Supersymmetric Gauge Theories

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with N _F charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(N _C) gauge theory with N _F hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(N _C) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.     

Moduli space of non-abelian vortices

We completely determine the moduli space M(N,k) of k vortices in U(N) gauge theory with N Higgs fields in the fundamental representation. Its open subset for separated vortices is found as the symmetric product (CxCP(N-1))k/(see text)k. Orbifold singularities of this space correspond to coincident vortices and are resolved resulting in a smooth moduli manifold. The relation to K?hler quotient construction is discussed.     

Sharp remainder of a critical Hardy inequality

An explicit representation is given to the remainder of a critical Hardy inequality in \({L^{n}(\mathbb{R}^{n})}\) with \({n \ge 2}\).     

Critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations

We consider a heat semigroup with an inverse square potential and prove critical dissipative estimates including the endpoint case. Some application to a nonlinear heat equation with an inverse square potential is also discussed.