CPAA
Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval
Marta Strani
Communications on Pure & Applied Analysis 2014, 13(4): 1653-1667 doi: 10.3934/cpaa.2014.13.1653
We study the existence and the uniqueness of a {\bf positive connection}, that is a stationary solution connecting the boundary data, for the initial-boundary value problem for the viscous shallow water system \begin{eqnarray} \partial_t u + \partial_x v=0, \partial_t v+\partial_x( \frac{v^2}{u}+ P(u))= \varepsilon\partial_x ( u \partial_x(\frac{v}{u})) \end{eqnarray} in a bounded interval $(-l,l)$ of the real line. We firstly consider the general case where the term of pressure $P(u)$ satisfies \begin{eqnarray} P(0)=0, P(+\infty)=+\infty, P'(u) \quad and \quad P''(u)>0 \ \forall u >0, \end{eqnarray} and then we show properties of the steady state in the relevant case $P(u)=\kappa u^{\gamma}$, $\gamma>1$. The viscous Saint-Venant system, corresponding to $\gamma=2$, fits in the general framework.
keywords: Saint-Venant system hyperbolic-parabolic systems stationary solutions Shallow water equations positive connection.
DCDS-S
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
$2 ≤ q < p$
and
$f$
is a nonlinearity of Fisher type on
$[0, 1]$
, namely
$f(0) = 0 = f(1)$
and
$f > 0$
on
$]0, 1[\, $
. We give an estimate of the critical speed and we comment on the roles of
$p$
and
$q$
in the dynamics, providing some numerical simulations.
keywords: Reaction-diffusion equations $(p, q)$-Laplacian wave fronts admissible speeds

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