Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential

2018 ◽  
Vol 15 (04) ◽  
pp. 755-788
Author(s):  
Vladimir Georgiev ◽  
Sandra Lucente
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Ground State ◽  
Nonlinear Term ◽  
Blow Up ◽  
Critical Energy ◽  
Ground State Solution ◽  
Energy Space ◽  
Nirenberg Inequality ◽  
Klein Gordon

We study the dynamics for the focusing nonlinear Klein–Gordon equation, [Formula: see text] with positive radial potential [Formula: see text] and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo–Nirenberg inequality gives a critical exponent depending on [Formula: see text]. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.

scholarly journals CROSS-CONSTRAINED VARIATIONAL PROBLEM AND THE NON-LINEAR KLEIN–GORDON EQUATIONS

2008 ◽  
Vol 50 (3) ◽  
pp. 467-481 ◽  
Author(s):  
ZAIHUI GAN ◽  
JIAN ZHANG
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Variational Method ◽  
Variational Problem ◽  
Invariant Manifolds ◽  
Invariant Sets ◽  
Sharp Threshold ◽  
Non Linear ◽  
Klein Gordon ◽  
Three Space

AbstractIn this paper, we put forward a cross-constrained variational method to study the non-linear Klein–Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.

scholarly journals RD-systems with chemotaxis: global existence, boundedness and blow-up of solutions

2021 ◽  
Author(s):  
Xue Xu ◽  
Zhong Huang
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Bootstrap Method ◽  
Critical Value ◽  
Iteration Procedure ◽  
Moser Iteration ◽  
Nirenberg Inequality

result shows that the blow-up is equivalent to the blow-up of the $L^r-$norms of the solutions for $r$ exceeding some critical value $r_c.$ Under very loose conditions we give the estimation of $r_c,$ relying on a variant of Gagliardo-Nirenberg inequality, and a kind of bootstrap method which is very similar to the Alikakos-Moser iteration procedure.

scholarly journals A system of Schrödinger equations with general quadratic-type nonlinearities

2020 ◽  
pp. 2050023
Author(s):  
Norman Noguera ◽  
Ademir Pastor
Keyword(s):  
Global Existence ◽  
Variational Methods ◽  
Nonlinear Stability ◽  
Ground State ◽  
Finite Time ◽  
Blow Up ◽  
Quadratic Growth ◽  
Schrodinger Equations ◽  
Concentration Compactness ◽  
Schrödinger Equations

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

scholarly journals Global Existence and Blow-Up of Solutions for Nonlinear Klein-Gordon Equation with Damping Term and Nonnegative Potentials

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Wen-Yi Huang ◽  
Wen-Li Chen
Keyword(s):  
Global Existence ◽  
Gordon Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Invariant Sets ◽  
Potential Well ◽  
Damping Term ◽  
Potential Wells ◽  
Klein Gordon ◽  
Klein Gordon Equation

This paper is concerned with the nonlinear Klein-Gordon equation with damping term and nonnegative potentials. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we obtain a new existence theorem of global solutions and a blow-up result for solutions in finite time.

Ground state solutions for nonlinear fractional Kirchhoff–Schrödinger–Poisson systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Li Wang ◽  
Tao Han ◽  
Kun Cheng ◽  
Jixiu Wang
Keyword(s):  
Growth Condition ◽  
Ground State ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Nonlocal Term ◽  
Ground State Solutions ◽  
Cerami Sequences ◽  
The Nehari Manifold ◽  
Fractional Laplace Operator

AbstractIn this paper, we study the existence of ground state solutions for the following fractional Kirchhoff–Schrödinger–Poisson systems with general nonlinearities:$$\left\{\begin{array}{ll}\left(a+b{\left[u\right]}_{s}^{2}\right)\,{\left(-{\Delta}\right)}^{s}u+u+\phi \left(x\right)u=\left({\vert x\vert }^{-\mu }\ast F\left(u\right)\right)f\left(u\right)\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \\ {\left(-{\Delta}\right)}^{t}\phi \left(x\right)={u}^{2}\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \end{array}\right.$$where$${\left[u\right]}_{s}^{2}={\int }_{{\mathrm{&#x211d;}}^{3}}{\vert {\left(-{\Delta}\right)}^{\frac{s}{2}}u\vert }^{2}\,\mathrm{d}x={\iint }_{{\mathrm{&#x211d;}}^{3}{\times}{\mathrm{&#x211d;}}^{3}}\frac{{\vert u\left(x\right)-u\left(y\right)\vert }^{2}}{{\vert x-y\vert }^{3+2s}}\,\mathrm{d}x\mathrm{d}y\,\text{,}$$$s,t\in \left(0,1\right)$ with $2t+4s{ >}3,0{< }\mu {< }3-2t,$$f:{\mathrm{&#x211d;}}^{3}{\times}\mathrm{&#x211d;}\to \mathrm{&#x211d;}$ satisfies a Carathéodory condition and (−Δ)s is the fractional Laplace operator. There are two novelties of the present paper. First, the nonlocal term in the equation sets an obstacle that the bounded Cerami sequences could not converge. Second, the nonlinear term f does not satisfy the Ambrosetti–Rabinowitz growth condition and monotony assumption. Thus, the Nehari manifold method does not work anymore in our setting. In order to overcome these difficulties, we use the Pohozǎev type manifold to obtain the existence of ground state solution of Pohozǎev type for the above system.

scholarly journals Ground State Solution of Pohožaev Type for Quasilinear Schrödinger Equation Involving Critical Exponent in Orlicz Space

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 779 ◽  
Author(s):  
Jianqing Chen ◽  
Qian Zhang
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Orlicz Space ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
Existence Results ◽  
Quasilinear Schrödinger Equation ◽  
Ground State Solutions

We study the following quasilinear Schrödinger equation involving critical exponent - Δ u + V ( x ) u - Δ ( u 2 ) u = A ( x ) | u | p - 1 u + λ B ( x ) u 3 N + 2 N - 2 , u ( x ) > 0 for x ∈ R N and u ( x ) → 0 as | x | → ∞ . By using a monotonicity trick and global compactness lemma, we prove the existence of positive ground state solutions of Pohožaev type. The nonlinear term | u | p - 1 u for the well-studied case p ∈ [ 3 , 3 N + 2 N - 2 ) , and the less-studied case p ∈ [ 2 , 3 ) , and for the latter case few existence results are available in the literature. Our results generalize partial previous works.

scholarly journals Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data

2008 ◽  
Vol 244 (11) ◽  
pp. 2693-2740 ◽  
Author(s):  
J. Emile Rakotoson ◽  
J. Michel Rakotoson ◽  
Cédric Verbeke
Keyword(s):  
Thin Films ◽  
Initial Data ◽  
Global Existence ◽  
Blow Up ◽  
Higher Order
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