scholarly journals Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data

2013 ◽  
Keyword(s):  
Liouville Equation ◽  
Blow Up ◽  
Blowing Up ◽  
Singular Data ◽  
Blowing Up Solutions ◽  
Morse Indices

scholarly journals Systems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Terms

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mauricio Bogoya ◽  
Julio D. Rossi
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Asymptotic Bounds ◽  
Blowing Up ◽  
Blow Up Rate ◽  
Blowing Up Solutions

We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

scholarly journals Explicit Blowing Up Solutions for a Higher Order Parabolic Equation with Hessian Nonlinearity

2021 ◽  
Author(s):  
Carlos Escudero
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Higher Order ◽  
Order Partial Differential Equation ◽  
Infinite Time ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Blowing Up ◽  
Blowing Up Solutions

AbstractIn this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite time, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions.

scholarly journals Blow up Points and the Morse Indices of Solutions to the Liouville Equation in Two-Dimension

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Futoshi Takahashi
Keyword(s):  
Bounded Domain ◽  
Liouville Equation ◽  
Morse Index ◽  
Blow Up ◽  
Smooth Bounded Domain ◽  
Two Dimension ◽  
Morse Indices

AbstractWe consider the Liouville equation−Δu = λeu in Ω, u =0 on∂Ω,on a smooth bounded domain Ω in ℝfor m ∈ℕ. We prove that the number of blow up points m is less than or equal to the Morse index of uAs a corollary, we show that if a solution u

scholarly journals Strong Instability of Ground States to a Fourth Order Schrödinger Equation

2017 ◽  
Vol 2019 (17) ◽  
pp. 5299-5315 ◽  
Author(s):  
Denis Bonheure ◽  
Jean-Baptiste Castéras ◽  
Tianxiang Gou ◽  
Louis Jeanjean
Keyword(s):  
Fourth Order ◽  
Ground State ◽  
Blow Up ◽  
Ground States ◽  
Rigorous Results ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Strong Instability ◽  
Blowing Up ◽  
Blowing Up Solutions

Abstract In this note, we prove the instability by blow-up of the ground state solutions for a class of fourth order Schrödinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic nonlinear Schrödinger due to Boulenger and Lenzmann [8] and confirm numerical conjectures from [1–3, 11].

scholarly journals Blowing-up solutions for a supercritical elliptic equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yessine Dammak
Keyword(s):  
Blow Up ◽  
Positive Function ◽  
Dimensional System ◽  
Positive Real ◽  
Subcritical Case ◽  
Smooth Bounded Domain ◽  
Finite Dimensional ◽  
Blowing Up ◽  
Two Bubbles ◽  
Blowing Up Solutions

<p style='text-indent:20px;'>This paper concerns the existence of solutions of the following supercritical PDE: <inline-formula><tex-math id="M1">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M2">\begin{document}$ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M7">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> positive function and <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of <inline-formula><tex-math id="M9">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula> having the form of two bubbles with non comparable speeds and which have only one blow-up point in <inline-formula><tex-math id="M10">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.</p>

Blow-up set of type I blowing up solutions for nonlinear parabolic systems

2016 ◽  
Vol 369 (3-4) ◽  
pp. 1491-1525 ◽  
Author(s):  
Yohei Fujishima ◽  
Kazuhiro Ishige ◽  
Hiroki Maekawa
Keyword(s):  
Blow Up ◽  
Parabolic Systems ◽  
Type I ◽  
Nonlinear Parabolic Systems ◽  
Nonlinear Parabolic ◽  
Blowing Up ◽  
Blowing Up Solutions
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