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1078-0947

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## Discrete and Continuous Dynamical Systems

January 2013 , Volume 33 , Issue 1

Special Issue

Tribute to Jean Mawhin

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2013, 33(1): i-ii
doi: 10.3934/dcds.2013.33.1i

*+*[Abstract](2306)*+*[PDF](98.2KB)**Abstract:**

Jean Mawhin will celebrate his seventieth birthday on December 11, 2012, most likely in Heusy (Verviers), a lovely city in the foothills of the Ardennes, in Belgium, where he was born and resided all his life. He received his pre-university education there and then continued with his studies of mathematics at the Université de Liège, where he received the degree of

*Docteur en Sciences Mathématiques, avec la plus grande distinction*on February 10, 1969. He continued as

*Maítre de Conférences*at Liège from 1969 until 1973 and simultaneously was appointed as

*Chargé de Cours*at the (newly established) Universitè Catholique de Louvain at Louvain-la-Neuve, from 1970 to 1974, where he then served as

*Professeur*from 1974 to 1977 and was promoted to

*Professeur Ordinaire*in 1977.

For more information please click the “Full Text” above.

2013, 33(1): 1-6
doi: 10.3934/dcds.2013.33.1

*+*[Abstract](2876)*+*[PDF](267.5KB)**Abstract:**

We consider Kolmogorov-type systems which are not necessarily competitive or cooperative. Our main result shows that such systems cannot have nontrivial periodic solutions whose orbits are orbitally stable. We obtain our results under two assumptions that we consider to be natural assumptions.

2013, 33(1): 7-26
doi: 10.3934/dcds.2013.33.7

*+*[Abstract](3566)*+*[PDF](462.4KB)**Abstract:**

We are concerned with the existence of single- and multi-bump solutions of the equation $-\Delta u+(\lambda a(x)+a_0(x))u=|u|^{p-2}u$, $x\in{\mathbb R}^N$; here $p>2$, and $p<\frac{2N}{N-2}$ if $N\geq 3$. We require that $a\geq 0$ is in $L^\infty_{loc}({\mathbb R}^N)$ and has a bounded potential well $\Omega$, i.e. $a(x)=0$ for $x\in\Omega$ and $a(x)>0$ for $x\in{\mathbb R}^N$\$\bar{\Omega}$. Unlike most other papers on this problem we allow that $a_0\in L^\infty({\mathbb R}^N)$ changes sign. Using variational methods we prove the existence of multibump solutions $u_\lambda$ which localize, as $\lambda\to\infty$, near prescribed isolated open subsets $\Omega_1,\dots,\Omega_k\subset\Omega$. The operator $L_0:=-\Delta+a_0$ may have negative eigenvalues in $\Omega_j$, each bump of $u_\lambda$ may be sign-changing.

2013, 33(1): 27-46
doi: 10.3934/dcds.2013.33.27

*+*[Abstract](2819)*+*[PDF](425.0KB)**Abstract:**

We investigate general properties, such as existence and uniqueness, continuous dependence on data and continuation, of solutions to retarded functional differential equations with infinite delay on a differentiable manifold.

2013, 33(1): 47-66
doi: 10.3934/dcds.2013.33.47

*+*[Abstract](2762)*+*[PDF](437.4KB)**Abstract:**

We deal with a class of functionals $I$ on a Banach space $X,$ having the structure $I=\Psi+\mathcal G,$ with $\Psi : X \to (- \infty , + \infty ]$ proper, convex, lower semicontinuous and $\mathcal G: X \to \mathbb{R} $ of class $C^1.$ Also, $I$ is $G$-invariant with respect to a discrete subgroup $G\subset X$ with $\mbox{dim (span}\ G)=N$. Under some appropriate additional assumptions we prove that $I$ has at least $N+1$ critical orbits. As a consequence, we obtain that the periodically perturbed $N$-dimensional relativistic pendulum equation has at least $N+1$ geometrically distinct periodic solutions.

2013, 33(1): 67-87
doi: 10.3934/dcds.2013.33.67

*+*[Abstract](2879)*+*[PDF](498.4KB)**Abstract:**

We extend the study [1] of gradient systems with equal depth multiple-well potentials to the case when some of the wells are degenerate, in the sense that the Hessian is non positive at those wells. The exponentially small speed, in terms of distances between fronts, typical of non degenerate potentials is replaced by an algebraic upper bound, whose degree depends on the degeneracy of the wells.

2013, 33(1): 89-110
doi: 10.3934/dcds.2013.33.89

*+*[Abstract](3935)*+*[PDF](472.5KB)**Abstract:**

We study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.

2013, 33(1): 111-122
doi: 10.3934/dcds.2013.33.111

*+*[Abstract](3138)*+*[PDF](362.4KB)**Abstract:**

This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \ 1 \leq p \leq +\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\Bbb{R}^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.

2013, 33(1): 123-140
doi: 10.3934/dcds.2013.33.123

*+*[Abstract](3030)*+*[PDF](441.7KB)**Abstract:**

sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish

*a priori*bounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties.

2013, 33(1): 141-152
doi: 10.3934/dcds.2013.33.141

*+*[Abstract](2977)*+*[PDF](374.6KB)**Abstract:**

In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$

2013, 33(1): 153-161
doi: 10.3934/dcds.2013.33.153

*+*[Abstract](2782)*+*[PDF](337.7KB)**Abstract:**

We prove results on when nonlinear elliptic equations have infinitely many bifurcations if the nonlinearities grow rapidly.

2013, 33(1): 163-172
doi: 10.3934/dcds.2013.33.163

*+*[Abstract](3025)*+*[PDF](402.9KB)**Abstract:**

In this paper we study critical points of the functional \begin{eqnarray*} J_{\epsilon}(u):= \frac{\epsilon^p}{p}\int_0^1|u_x|^pdx+\int_0^1F(u)dx, \; u∈w^{1,p}(0,1), \end{eqnarray*} where F:$\mathbb{R}$→$\mathbb{R}$ is assumed to be a double-well potential. This functional represents the total free energy in phase transition models. We consider a non-classical choice for $F$ modeled on $F(u)=|1-u^2|^{\alpha}$ where $1< \alpha < p$. This choice leads to the existence of multiple continua of critical points that are not present in the classical case $\alpha= p = 2$. We prove that the interior of these continua are local minimizers. The energy of these local minimizers is strictly greater than the global minimum of $J_{\epsilon}$. In particular, the existence of these continua suggests an alternative explanation for the

*slow dynamics*observed in phase transition models.

2013, 33(1): 173-191
doi: 10.3934/dcds.2013.33.173

*+*[Abstract](3824)*+*[PDF](451.4KB)**Abstract:**

We study positive solutions $y(u)$ for the first order differential equation $$y'=q(c\,{y}^{\frac{1}{p}}-f(u))$$ where $c>0$ is a parameter, $p>1$ and $q>1$ are conjugate numbers and $f$ is a continuous function in $[0,1]$ such that $f(0)=0=f(1)$. We shall be particularly concerned with positive solutions $y(u)$ such that $y(0)=0=y(1)$. Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the $p$-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for $p=2$.

2013, 33(1): 193-210
doi: 10.3934/dcds.2013.33.193

*+*[Abstract](3270)*+*[PDF](436.7KB)**Abstract:**

Yakubovich, Fradkov, Hill and Proskurnikov have used the Yaku-bovich Frequency Theorem to prove that a strictly dissipative linear-quadratic control process with periodic coefficients admits a storage function, and various related results. We extend their analysis to the case when the coefficients are bounded uniformly continuous functions.

2013, 33(1): 211-223
doi: 10.3934/dcds.2013.33.211

*+*[Abstract](3463)*+*[PDF](125.4KB)**Abstract:**

By using differential inequalities we improve some estimates of W.S. LOUD for the ultimate bound and asymptotic stability of the solutions to the Duffing equation $ u''+ c{u'} + g(u)= f(t)$ where $c>0$, $f $ is measurable and essentially bounded, and $g$ is continuously differentiable with $g'\ge b>0$.

2013, 33(1): 225-237
doi: 10.3934/dcds.2013.33.225

*+*[Abstract](3249)*+*[PDF](371.5KB)**Abstract:**

This paper is concerned with non-selfadjoint elliptic problems having a principal part in divergence form and involving an indefinite weight. We study the asymptotic behavior of the principal eigenvalues when the first order term (drift term) becomes larger and larger. Several of our results also apply to elliptic operators in general form.

2013, 33(1): 239-254
doi: 10.3934/dcds.2013.33.239

*+*[Abstract](2610)*+*[PDF](461.2KB)**Abstract:**

We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations $L(p)x=f(x)+b(t),\;p=d/dt$ with $2\pi$-periodic forcing $b$ and periodic $f$ we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of $2\pi$-periodic solutions.

2013, 33(1): 255-276
doi: 10.3934/dcds.2013.33.255

*+*[Abstract](2503)*+*[PDF](496.1KB)**Abstract:**

This paper is about systems of variational inequalities of the form: $$ \left\{ \begin{array}{l} ‹A_k U_k+ F_k (u) , v_k -u_k› ≥ 0,\; ∀ v_k ∈ K_k \\ u_k ∈ K_k , \end{array} \right. $$ $(k=1,\dots , m)$, where $A_k$ and $F_k$ are multivalued mappings with possibly non-power growths and $K_k$ is a closed, convex set. We concentrate on the noncoercive case and follow a sub-supersolution approach to obtain the existence and enclosure of solutions to the above system between sub- and supersolutions.

2013, 33(1): 277-282
doi: 10.3934/dcds.2013.33.277

*+*[Abstract](3462)*+*[PDF](288.9KB)**Abstract:**

In this work we study the periodic solutions, their stability and bifurcation for the class of Duffing differential equation $x''+ \epsilon C x'+ \epsilon^2 A(t) x +b(t) x^3 = \epsilon^3 \Lambda h(t)$, where $C>0$, $\epsilon>0$ and $\Lambda$ are real parameter, $A(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions and $\epsilon$ is sufficiently small. Our results are proved using the averaging method of first order.

2013, 33(1): 283-303
doi: 10.3934/dcds.2013.33.283

*+*[Abstract](3328)*+*[PDF](490.5KB)**Abstract:**

This paper deals with integral equations of the form \begin{eqnarray*} x(t)=\tilde{x}+∫_a^td[A]x+f(t)-f(a), t∈[a,b], \end{eqnarray*} in a Banach space $X,$ where $-\infty\ < a < b < \infty$, $\tilde{x}∈ X,$ $f:[a,b]→X$ is regulated on [a,b] and $A(t)$ is for each $t∈[a,b], $ a linear bounded operator on $X,$ while the mapping $A:[a,b]→L(X)$ has a bounded variation on [a,b] Such equations are called generalized linear differential equations. Our aim is to present new results on the continuous dependence of solutions of such equations on a parameter. Furthermore, an application of these results to dynamic equations on time scales is given.

2013, 33(1): 305-320
doi: 10.3934/dcds.2013.33.305

*+*[Abstract](2925)*+*[PDF](410.6KB)**Abstract:**

We discuss existence and regularity of bounded variation solutions of the Dirichlet problem for the one-dimensional capillarity-type equation \begin{equation*} \Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = f(t,u) \quad \hbox{ in } {]-r,r[}, \qquad u(-r)=a, \, u(r) = b. \end{equation*} We prove interior regularity of solutions and we obtain a precise description of their boundary behaviour. This is achieved by a direct and elementary approach that exploits the properties of the zero set of the right-hand side $f$ of the equation.

2013, 33(1): 321-333
doi: 10.3934/dcds.2013.33.321

*+*[Abstract](3203)*+*[PDF](424.0KB)**Abstract:**

We prove the existence of an unbounded sequence of critical points of the functional \begin{equation*} J_{\lambda}(u) =\frac{1} {p} ∫_{\mathbb{R} ^N}{||x|^{-α\nabla^k} u|} ^p - λ h(x){||x|^{-α+k}u|} ^p - \frac{1} {q} ∫_{\mathbb{R} ^N}Q(x){||x|^{-b}u|} ^q \end{equation*} related to the Caffarelli-Kohn-Nirenberg inequality and its higher order variant by Lin. We assume $Q\le 0$ at 0 and infinity and consider two essentially different cases: $h\equiv 1$ and $h$ in a certain weighted Lebesgue space.

2013, 33(1): 335-344
doi: 10.3934/dcds.2013.33.335

*+*[Abstract](3195)*+*[PDF](389.0KB)**Abstract:**

Consider the following nonlinear elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u - u=\mu_1u^3+\beta uv^2,\ & \hbox{in}\ \Omega\\ -\Delta v - v= \mu_2v^3+\beta vu^2,\ & \hbox{in}\ \Omega\\ u,v>0\ \hbox{in}\ \Omega, \ u=v=0,\ & \hbox{on}\ \partial\Omega, \end{array} \right. \end{equation*}where $\mu_1,\mu_2>0$ are constants and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\leq3$. We study the existence and non-existence of positive solutions and give bifurcation results in terms of the coupling constant $\beta$.

2013, 33(1): 345-357
doi: 10.3934/dcds.2013.33.345

*+*[Abstract](2507)*+*[PDF](456.7KB)**Abstract:**

We consider second-order Euler-Lagrange systems which are periodic in time. Their periodic solutions may be characterized as the stationary points of an associated action functional, and we study the dynamical implications of minimizing the action. Examples are well-known of stable periodic minimizers, but instability always holds for periodic solutions which are minimal in the sense of Aubry-Mather.

2013, 33(1): 359-363
doi: 10.3934/dcds.2013.33.359

*+*[Abstract](2660)*+*[PDF](264.5KB)**Abstract:**

In this paper we discuss some general results on families of symmetric and doubly-symmetric solutions in reversible Hamiltonian systems having several independent first integrals. We describe a set-up for such solutions which allows the application of classical continuation and bifurcation results.

2013, 33(1): 365-379
doi: 10.3934/dcds.2013.33.365

*+*[Abstract](3537)*+*[PDF](349.4KB)**Abstract:**

In 1806 Poisson published one of the first papers on functional differential equations. Among others he studied an example with a state-dependent delay, which is motivated by a geometric problem. This example is not covered by recent results on initial value problems for differential equations with state-dependent delay. We show that the example generates a semiflow of differentiable solution operators, on a manifold of differentiable functions and away from a singular set. Initial data in the singular set produce multiple solutions.

2013, 33(1): 381-389
doi: 10.3934/dcds.2013.33.381

*+*[Abstract](2850)*+*[PDF](322.6KB)**Abstract:**

Let $f\in C(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m},% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m})$ and $p\in C([0,T],% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m})$ be continuous functions. We consider the $T$ periodic boundary value problem (*) $u^{\prime}(t)=f(u(t))+p(t),$ $u(0)=u(T).$ It is shown that when $f$ is a coercive gradient function, or the bounded perturbation of a coercive gradient function, and the Brouwer degree $d_{B}(f,B(0,r),0)\neq0$ for large $r$, there is a solution for all $p.$ A result for bounded $f$ is also obtained.

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