Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^2$-critical. On the other...

Let $L(z,{\partial }_z)=({\partial }_{z_0}{)}^k-A(z,{\partial }_z)$ be a linear partial differential operator with holomorphic coefficients, where$$A(z,{\partial }_z)=\sum _{j=0}^{k-1}{A}_j(z,{\partial }_{z^{\text{'}}}\left)\right({\partial }_{z_0}{)}^j,\mathrm{ord}.A(z,{\partial }_z)=m\>k$$and$$z=(z_0,z^{\text{'}})\in C^{n+1}.$$We consider Cauchy problem with holomorphic data$$L(z,{\partial }_z)u\left(z\right)=f\left(z\right),\left({\partial }_{z_0}{)}^{i}u\right(0,z^{\text{'}})={\widehat{u}}_i(z^{\text{'}}\left)\right(0\le i\le k-1).$$We can easily get a formal solution $\widehat{u}\left(z\right)={\sum }_{n=0}^{\infty }{\widehat{u}}_n(z^{\text{'}}\left)\right(z_0{)}^n/n!$, bu in general it diverges. We show under some conditions that for any sector $S$ with the opening less that a constant determined by $L(z,{\partial }_z)$, there is a function $u_S\left(z\right)$ holomorphic except on $\{z_0=0\}$ such that $L(z,{\partial }_z)u_S\left(z\right)=f\left(z\right)$ and $u_S\left(z\right)\sim \widehat{u}\left(z\right)$ as $z_0\to 0$ in $S$.

Let $T$ be a positive number or $+\infty$. We characterize all subsets $M$ of ${ℝ}^n\times ]0,T[$ such that $$\underset{X\in {ℝ}^n\times ]0,T[}{inf}u\left(X\right)=\underset{X\in M}{inf}u\left(X\right)i$$ for every positive parabolic function $u$ on ${ℝ}^n\times ]0,T[$ in terms of coparabolic (minimal) thinness of the set $M_{\delta }={\cup }_{(x,t)\in M}{B}^p((x,t),\delta t)$, where $\delta \in (0,1)$ and $B^p((x,t),r)$ is the “heat ball” with the “center” $(x,t)$ and radius $r$. Examples of different types of sets which can be used instead of “heat balls” are given. It is proved that (i) is equivalent to the condition ${sup}_{X\in {ℝ}^n\times {ℝ}^{+}}u\left(X\right)={sup}_{X\in M}u\left(X\right)$ for every bounded parabolic function on ${ℝ}^n\times {ℝ}^{+}$ and hence to all equivalent conditions given in the article [7]....

The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawłucki's extension of the Puiseux lemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.

The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $$u_t=-A\left(t\right)u_{x^4}+B\left(t\right)u_{x^2}+g{\left(u\right)}_{x^2}+f{\left(u\right)}_x+h{\left(u_x\right)}_x+G\left(u\right)$$ with the initial boundary value conditions $$u(-\ell ,t)=u(\ell ,t)=0,u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ or with the initial boundary value conditions $$u_x(-\ell ,t)=u_x(\ell ,t)=0,u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...