scholarly journals Anisotropic problems with unbalanced growth

2020 ◽  
Vol 9 (1) ◽  
pp. 1504-1515
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Variational Methods ◽  
General Class ◽  
Entire Solution ◽  
Entire Solutions ◽  
Power Type ◽  
Unbalanced Growth ◽  
Lack Of Compactness ◽  
Anisotropic Problems

Abstract The main purpose of this paper is to study a general class of (p, q)-type eigenvalues problems with lack of compactness. The reaction is a convex-concave nonlinearity described by power-type terms. Our main result establishes a complete description of all situations that can occur. We prove the existence of a critical positive value λ* such that the following properties hold: (i) the problem does not have any entire solution in the case of low perturbations (that is, if 0 < λ < λ*); (ii) there is at least one solution if λ = λ*; and (iii) the problem has at least two entire solutions in the case of high perturbations (that is, if λ > λ*). The proof combines variational methods, analytic tools, and monotonicity arguments.

ENTIRE SOLUTIONS OF A CURVATURE FLOW IN AN UNDULATING CYLINDER

2018 ◽  
Vol 99 (1) ◽  
pp. 137-147
Author(s):  
LIXIA YUAN ◽  
BENDONG LOU
Keyword(s):  
Curvature Flow ◽  
Entire Solution ◽  
Travelling Wave ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Entire Solutions ◽  
Smooth Functions ◽  
Periodic Traveling Waves ◽  
Periodic Travelling Wave ◽  
Positive Functions

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y)<x<g_{2}(y),y\in \mathbb{R}\}$, where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. [‘Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit’, Netw. Heterog. Media1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$.

scholarly journals Entire solutions of the Fisher–KPP equation on the half line

2019 ◽  
Vol 31 (3) ◽  
pp. 407-422 ◽  
Author(s):  
BENDONG LOU ◽  
JUNFAN LU ◽  
YOSHIHISA MORITA
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Traveling Wave ◽  
Wave Solution ◽  
Dirichlet Boundary ◽  
Entire Solution ◽  
Traveling Wave Solution ◽  
Half Line ◽  
Entire Solutions ◽  
Kpp Equation

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

scholarly journals Measures of Growth of Entire Solutions of Generalized Axially Symmetric Helmholtz Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Devendra Kumar ◽  
Rajbir Singh
Keyword(s):  
Helmholtz Equation ◽  
Lower Order ◽  
Entire Solution ◽  
Axially Symmetric ◽  
Entire Solutions ◽  
Lower Type ◽  
Order And Type

For an entire solution of the generalized axially symmetric Helmholtz equation , measures of growth such as lower order and lower type are obtained in terms of the Bessel-Gegenbauer coefficients. Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients.

scholarly journals Entire Solutions of an Integral Equation in R5

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Xin Feng ◽  
Xingwang Xu
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Integral Form ◽  
Entire Solution ◽  
Entire Solutions ◽  
Positive Entire Solution ◽  
Moving Sphere Method

We will study the entire positive C0 solution of the geometrically and analytically interesting integral equation: u(x)=1/C5∫R5‍|x-y|u-q(y)dy with 0<q in R5. We will show that only when q=11, there are positive entire solutions which are given by the closed form u(x)=c(1+|x|2)1/2 up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that q must be equal to 11 if there exists a positive entire solution to the integral equation. The tool to reach this conclusion is the well-known Pohozev identity. The amazing cancelation occurred in Pohozev’s identity helps us to conclude the claim. It is this exponent which makes the moving sphere method work. In the second part, as normal, we adopt the moving sphere method based on the integral form to solve the integral equation.

scholarly journals On entire solutions of a certain type of nonlinear differential equation

2001 ◽  
Vol 64 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Chung-Chun Yang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Finite Order ◽  
Nonlinear Differential Equation ◽  
Entire Solution ◽  
Differential Polynomial ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential Polynomial ◽  
Linear Differential

In this note, we shall study, via Nevanlinna's value distribution theory, the uniqueness of transcendental entire solutions of the following type of nonlinear differential equation: (*) L (f (z)) – p (z) fn(z) = h (z), where L (f) denotes a linear differential polynomial in f with polynomials as its co-efficients, p (z) a polynomial (≢ 0), h an entire function, and n an integer ≥ 3. We show that if the equation (*) has a finite order transcendental entire solution, then it must be unique, unless L (f) ≡ 0.

Entire solutions for some reaction–diffusion systems

2015 ◽  
Vol 08 (04) ◽  
pp. 1550052 ◽  
Author(s):  
Guangying Lv ◽  
Dang Luo
Keyword(s):  
Classical Solution ◽  
General Model ◽  
Reaction Diffusion ◽  
Entire Solution ◽  
Reaction Model ◽  
Entire Solutions ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Time Variables ◽  
Belousov Zhabotinskii Reaction

This paper is concerned with the existence of entire solutions of some reaction–diffusion systems. We first consider Belousov–Zhabotinskii reaction model. Then we study a general model. Using the comparing argument and sub-super-solutions method, we obtain the existence of entire solutions which behave as two wavefronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. At last, we give some examples to explain our results for the general models.

scholarly journals Solitary waves for a fractional Klein–Gordon–Maxwell equations

2021 ◽  
pp. 1-13
Author(s):  
Xin Zhang
Keyword(s):  
Variational Methods ◽  
Solitary Waves ◽  
Symmetric Solution ◽  
Maxwell Equations ◽  
Existence Of Solutions ◽  
Critical System ◽  
Radially Symmetric Solution ◽  
Lack Of Compactness ◽  
Maxwell’S Equation ◽  
Klein Gordon

We investigate existence of solutions for a fractional Klein–Gordon coupled with Maxwell's equation. On the basis of overcoming the lack of compactness, we obtain that there is a radially symmetric solution for the critical system by means of variational methods.

NONAUTONOMOUS BIFURCATION OF BOUNDED SOLUTIONS: CROSSING CURVE SITUATIONS

2012 ◽  
Vol 12 (02) ◽  
pp. 1150017 ◽  
Author(s):  
CHRISTIAN PÖTZSCHE
Keyword(s):  
Differential Equations ◽  
Variational Equation ◽  
Entire Solution ◽  
Nondegeneracy Condition ◽  
Entire Solutions ◽  
Fold Bifurcation ◽  
Nonautonomous Differential Equations ◽  
Random Differential Equations ◽  
Pitchfork Bifurcations ◽  
Time Invariant

Carathéodory differential equations naturally occur as path-wise realization of random differential equations and are amenable for deterministic calculus. In the setup of such nonautonomous differential equations with only measurable time-dependence, we present an approach to a bifurcation theory based on a topological change in the set of bounded entire solutions. In such a setting of at least planar equations, we provide sufficient criteria for a nonhyperbolic entire solution to bifurcate into two branches of bounded or homoclinic solutions. As opposed to transcritical or pitchfork bifurcations, no trivial solution branch is supposed to exist in advance. In particular, we discuss a degenerate fold bifurcation pattern, where the transversality assumption is replaced by a nondegeneracy condition on the second-order derivative. Both bifurcation patterns are intrinsically nonautonomous and do not occur for time-invariant equations. Our notion of a nonhyperbolic solution is based on the fact that the associate variational equation possesses exponential dichotomies on both semiaxes with compatible projectors. The resulting Fredholm theory allows one to apply recent abstract bifurcation results due to Liu, Shi and Wang (2007).

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