Existence of heteroclinic solution for a class of non-autonomous second-order equation

Alves, Claudianor O.

doi:10.1007/s00030-015-0319-0

Cited by 6 publications

(6 citation statements)

References 10 publications

(10 reference statements)

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“…Let B ⊂ X be a bounded subset, x ∈ B, and ρ 0 > 0 such that x X < ρ 0 . Consider m(t) given by (6) with Ω = B.…”

Section: Existence Resultsmentioning

confidence: 99%

“…The study of differential equations and boundary value problems on the half-line or in the whole real line and the existence of homoclinic or heteroclinic solutions have received increasing interest in the last few years, due to the applications to non-Newtonian fluids theory, the diffusion of flows in porous media, and nonlinear elasticity (see, for instance, [6][7][8][9][10][11][12][13][14][15][16] and the references therein). In particular, heteroclinic connections are related to processes in which the variable transits from an unstable equilibrium to a stable one (see, for example, [17][18][19][20][21][22][23][24]); that is why heteroclinic solutions are often called transitional solutions.…”

Section: Introductionmentioning

confidence: 99%

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Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

Minhós

2019

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In this paper, we consider the second order discontinuous differential equation in the real line, a t , u ϕ u ′ ′ = f t , u , u ′ , a . e . t ∈ R , u ( − ∞ ) = ν − , u ( + ∞ ) = ν + , with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a ∈ C ( R 2 , R ) with a ( t , x ) > 0 for ( t , x ) ∈ R 2 , f : R 3 → R a L 1 -Carathéodory function and ν − , ν + ∈ R such that ν − < ν + . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities ϕ and f . To the best of our knowledge, this result is even new when ϕ ( y ) = y , that is for equation a t , u ( t ) u ′ ( t ) ′ = f t , u ( t ) , u ′ ( t ) , a . e . t ∈ R . Moreover, these results can be applied to classical and singular ϕ -Laplacian equations and to the mean curvature operator.

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“…Let B ⊂ X be a bounded subset, x ∈ B, and ρ 0 > 0 such that x X < ρ 0 . Consider m(t) given by (6) with Ω = B.…”

Section: Existence Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

Minhós

2019

Axioms

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“…It may be plausible that the above theorem generalizes to the case where h ∞ is a periodic function (with the obvious interpretation of (3.1)). A related result for the scalar problem can be found in [9]. It is worth mentioning that the results in [25] do not require the corresponding inhomogeneity to be asymptotically periodic (however they require further assumptions on the corresponding potential which include the nondegeneracy of its global minima).…”

Section: )mentioning

confidence: 89%

“…To the best of our knowledge, there are only a few corresponding results for spatially inhomogeneous systems, which we will refer to in the subsequent remarks. For the state of the art in the case of the scalar problem, we refer the interested reader to [9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

The Heteroclinic Connection Problem for General Double-Well Potentials

Sourdis

2016

Mediterr. J. Math.

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“…More recently, BVPs on the half or the whole line have been considered with surjective or non‐surjective (singular) homeomorphisms, and sufficient conditions for the existence of homoclinic or heteroclinic solutions were obtained (see, for instance, and the references there in) or for the solvability of problems with integral boundary conditions (). In , it is studied the problem

\begin{matrix} true \\ right {((), ϕ, (u^{'} (t)))}^{'} & = & left f (() t, u false(t false), u^{'} false(t false)), on double-struckR, \\ right u false(- \infty false) & = & left - 1, u false(+ \infty false) = 1, \end{matrix}

with the following assumptions on the nonlinearity f :

false(f 0 false) :

f : {double-struckR}^{3} \to R

is continuous and satisfies the symmetry condition

f false(t, x, y false) = - f false(- t, - x, y false) for all false(t, x, y false) \in {double-struckR}^{3};

false(f 1 false) :

f (t, 1, y) = 0 = f (t, - 1, y)

for all t ,

y \in R

;

false(f 2 false) :

f

…”

Section: Introductionmentioning

confidence: 99%

On heteroclinic solutions for BVPs involving ‐Laplacian operators without asymptotic or growth assumptions

Minhós

2018

Mathematische Nachrichten

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In this paper we consider the second order discontinuous equation in the real line, trueright156.0ptright(ϕfalse(a(t)u′(t)false))′=leftffalse(t,u(t),u′(t)false),0.33ema.e.0.33emt∈double-struckR,rightufalse(−∞false)=leftA,0.16em0.16emufalse(+∞false)=B,with ϕ an increasing homeomorphism such that ϕ(0)=0 and ϕ(R)=R, a∈C(R) with a(t)>0, for t∈R, f:double-struckR3→R a L1‐Carathéodory function and A,B∈R verifying an adequate relation. We remark that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities ϕ and f. Moreover, as far as we know, our main result is even new when ϕ(y)=y, that is, for the equation (afalse(tfalse)u′false(tfalse))′=ffalse(t,u(t),u′(t)false),a.e.t∈R.

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Existence of heteroclinic solution for a class of non-autonomous second-order equation

Abstract: In this paper, we use variational methods to prove the existence of heteroclinic solutions for a class of non-autonomous second-order equation.

Cited by 6 publications

References 10 publications

Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

The Heteroclinic Connection Problem for General Double-Well Potentials

On heteroclinic solutions for BVPs involving ‐Laplacian operators without asymptotic or growth assumptions

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