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In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.
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We prove Strichartz estimates with a loss of derivatives for the Schrodinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a result of the second author regarding the Schrodinger equation on the Euclidean cone.
Let $u$ be the solution of $u_t=Deltalog u$ in $R^Ntimes (0,T)$, N=3 or $Nge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)le u_0le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-log (T-t)$, converges uniformly on $R^N$ to the rescaled Barenblatt solution $4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $stoinfty$. We also obtain convergence of the rescaled solution $4{u}(x,s)$ as $stoinfty$ when the initial data satisfies $0le u_0(x)le B_{k_0}(x,0)$ in $R^N$ and $|u_0(x)-B_{k_0}(x,0)|le f(|x|)in L^1(R^N)$ for some constant $k_0>0$ and some radially symmetric function $f$.
We consider the cubic Hyperbolic Schrodinger equation eqref{eq:nls} on torus $T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that eqref{eq:nls} is analytic locally well-posed in in $H^s(T^2)$ with $s>1/2$, meanwhile, the ill-posedness in $H^s(T^2)$ for $s<1/2$ is also obtained. The main difficulty comes from estimating the number of representations of an integer as a difference of squares.
The initial-boundary value problem (IBVP) for the nonlinear Schrodinger (NLS) equation on the half-plane with nonzero boundary data is studied by advancing a novel approach recently developed for the well-posedness of the cubic NLS on the half-line, which takes advantage of the solution formula produced by the unified transform of Fokas for the associated linear IBVP. For initial data in Sobolev spaces on the half-plane and boundary data in Bourgain spaces arising naturally when the linear IBVP is solved with zero initial data, the present work provides a local well-posedness result for NLS initial-boundary value problems in higher dimensions.
We consider the derivative nonlinear Schrodinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$-critical with respect to scaling. The first question we discuss is whether ensembles of orbits with $L^2$-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction $M(q)=int |q|^2 < 4pi$. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well-posed for initial data in $H^{1/6}$ under the same restriction on $M$. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.
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