Periodic solutions of the Brillouin electron beam focusing equation
Maurizio Garrione Manuel Zamora
Communications on Pure & Applied Analysis 2014, 13(2): 961-975 doi: 10.3934/cpaa.2014.13.961
Quite unexpectedly with respect to the numerical and analytical results found in literature, we establish a new range for the real parameter $b$ for which the existence of $2\pi-$periodic solutions of the Brillouin focusing beam equation \begin{eqnarray} \ddot{x}+b(1+\cos t)x=\frac{1}{x} \end{eqnarray} is guaranteed. This is possible thanks to suitable nonresonance conditions acting on the rotation number of the solutions in the phase plane.
keywords: Brillouin focusing system Poincaré-Bohl theorem singular nonlinearities periodic solutions
Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations
Maurizio Garrione Marta Strani
Discrete & Continuous Dynamical Systems - S 2019, 12(1): 91-103 doi: 10.3934/dcdss.2019006
We study the existence of monotone heteroclinic traveling waves for the
-dimensional reaction-diffusion equation
$u_t = (\vert u_x \vert^{p-2} u_x + \vert u_x \vert^{q-2} u_x)_x + f(u), \;\;\;\; t ∈ \mathbb{R}, \; x ∈ \mathbb{R}, $
where the non-homogeneous operator appearing on the right-hand side is of
$(p, q)$
-Laplacian type. Here we assume that
$2 ≤ q < p$
is a nonlinearity of Fisher type on
$[0, 1]$
, namely
$f(0) = 0 = f(1)$
$f > 0$
$]0, 1[\, $
. We give an estimate of the critical speed and we comment on the roles of
in the dynamics, providing some numerical simulations.
keywords: Reaction-diffusion equations $(p, q)$-Laplacian wave fronts admissible speeds
Positive solutions to indefinite Neumann problems when the weight has positive average
Alberto Boscaggin Maurizio Garrione
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5231-5244 doi: 10.3934/dcds.2016028
We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$ u'' + q(t)g(u) = 0, \quad t \in [0, T], $$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(u) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$ x' = y, \qquad y' = h(x)y^2 + q(t), $$ with $h(x)$ a continuous function defined on the whole real line.
keywords: average condition Neumann problem Indefinite weight shooting method.

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