We prove local existence and uniqueness for the Newton-Schroedinger equation in three dimensions. Further we show that the blow-up alternative holds true as well as the continuous dependence of the solution w.r.t. the initial data.
We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in a backward light-cone and approaches the ODE blowup profile.
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.
We revisit the following nonlinear critical elliptic equation begin{equation*} -Delta u+Q(y)u=u^{frac{N+2}{N-2}},;;; u>0;;;hbox{ in } mathbb{R}^N, end{equation*} where $Ngeq 5.$ Although there are some existence results of bubbling solutions for problem above, there are no results about the periodicity of bubbling solutions. Here we investigate some related problems. Assuming that $Q(y)$ is periodic in $y_1$ with period 1 and has a local minimum at 0 satisfying $Q(0)=0,$ we prove the existence and local uniqueness of infinitely many bubbling solutions of the problem above. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the potential function $Q(y),$ i.e. the bubbling solution whose blow-up set is ${(jL,0,...,0):j=0,1,2,...,m}$ must be periodic in $y_{1}$ provided that $L$ is large enough, where $m$ is the number of the bubbles which is large enough but independent of $L.$ Moreover, we also show a non-existence of this bubbling solutions for the problem above if the local minimum of $Q(y)$ does not equal to zero.
We get a local existence result in $H^s$ with $s>3/2$ for second order quasilinear wave equation with radial initial data in 2+1 dimensions, based on an improvement of Strichartz estimate in the radial case. Moreover, we get the corresponding local well-posed result for semilinear wave equation. The required index of regularity here is 1/4 less than the index 7/4, which is essentially sharp in general.
In this paper we prove the existence of vortices, namely standing waves with non null angular momentum, for the nonlinear Klein-Gordon equation in dimension $Ngeq 3$. We show with variational methods that the existence of these kind of solutions, that we have called emph{hylomorphic vortices}, depends on a suitable energy-charge ratio. Our variational approach turns out to be useful for numerical investigations as well. In particular, some results in dimension N=2 are reported, namely exemplificative vortex profiles by varying charge and angular momentum, together with relevant trends for vortex frequency and energy-charge ratio. The stability problem for hylomorphic vortices is also addressed. In the absence of conclusive analytical results, vortex evolution is numerically investigated: the obtained results suggest that, contrarily to solitons with null angular momentum, vortex are unstable.
Ali BenAmor
,Philippe Blanchard
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(2009)
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"Local existence and uniqueness for the frictional Newton-Schroedinger equation in three dimensions"
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Ali BenAmor
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