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### Open Access Journals

DCDS

In this paper we prove decay estimates for solutions to a nonlocal $p-$Laplacian evolution
problem with mixed boundary conditions, that is, $$u_t (x,t)=\int_{\Omega\cup\Omega_0} J(x-y) |u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))\, dy$$ for
$(x,t)\in \Omega\times \mathbb{R}^+$ and
$u(x,t)=0$ in $ \Omega_0\times \mathbb{R}^+$. The proof of these estimates is based on bounds for the associated
first eigenvalue.

CPAA

We consider the solution $u_p$ to the Neumann problem for the
$p$--Laplacian equation with the normal component of the flux
across the boundary given by $g\in L^\infty(\partial\Omega)$. We
study the behaviour of $u_p$ as $p$ goes to $1$ showing that they
converge to a measurable function $u$ and the gradients $|\nabla
u_p|^{p-2}\nabla u_p$ converge to a vector field $z$.
We prove that $z$ is bounded and that the properties of $u$ depend
on the size of $g$ measured in a suitable norm: if $g$ is small
enough, then $u$ is a function of bounded variation (it vanishes
on the whole domain, when $g$ is very small) while if $g$ is large
enough, then $u$ takes the value $\infty$ on a set of positive
measure. We also prove that in the first case, $u$ is a solution
to a limit problem that involves the $1-$Laplacian. Finally,
explicit examples are shown.

CPAA

In this paper we study the limit of Monge-Kantorovich mass
transfer problems when the involved measures are supported in a
small strip near the boundary of a bounded smooth domain,
$\Omega$. Given two absolutely continuos measures (with respect to
the surface measure) supported on the boundary $\partial
\Omega$, by performing a suitable extension of the measures to a
strip of width $\varepsilon$ near the boundary of the domain $\Omega$ we
consider the mass transfer problem for the extensions. Then we
study the limit as $\varepsilon$ goes to zero of the Kantorovich
potentials for the extensions and obtain that it coincides with a
solution of the original mass transfer problem. Moreover we look
for the possible approximations of these problems by solutions to
equations involving the $p-$Laplacian for large values of $p$.

DCDS

In this paper we prove a general result concerning continuity of
the blow-up time and the blow-up set for an evolution problem
under perturbations. This result is based on some convergence of
the perturbations for times smaller than the blow-up time of the
unperturbed problem together with uniform bounds on the blow-up
rates of the perturbed problems.

We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.

We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.

CPAA

In this paper we look for PDEs that arise as limits of values of
tug-of-war games when the possible movements of the game are taken
in a family of sets that are not necessarily Euclidean balls. In
this way we find existence of viscosity solutions to the Dirichlet
problem for an equation of the form
$- \langle D^2 v\cdot J_x(D v) ; J_x(Dv)\rangle (x) =0$,
that is, an infinity Laplacian with spatial dependence.
Here $J_x (Dv(x))$ is a vector that depends on
the spatial location and the gradient of the solution.

CPAA

We study the asymptotic behavior for the
best constant and extremals of the Sobolev trace embedding
$W^{1,p} (\Omega) \rightarrow L^q (\partial \Omega)$ on
expanding and contracting domains. We find that the behavior
strongly depends on $p$ and $q$. For contracting domains we prove
that the behavior of the best Sobolev trace constant depends on
the sign of $qN-pN+p$ while for expanding domains it depends on
the sign of $q-p$. We also give some results regarding the
behavior of the extremals, for contracting domains we prove that
they converge to a constant when rescaled in a suitable way and
for expanding domains we observe when a concentration phenomena
takes place.

CPAA

In this work we consider the nonlocal stationary
nonlinear problem $(J* u)(x) - u(x)= -\lambda u(x)+ a(x) u^p(x)$ in a
domain $\Omega$, with the Dirichlet boundary condition $u(x)=0$ in
$\mathbb{R}^N\setminus \Omega$ and $p>1$. The kernel $J$ involved in the
convolution $(J*u) (x) = \int_{\mathbb{R}^N} J(x-y) u(y) dy$ is a
smooth, compactly supported nonnegative function with unit
integral, while the weight $a(x)$ is assumed to be nonnegative and
is allowed to vanish in a smooth subdomain $\Omega_0$ of $\Omega$. Both
when $a(x)$ is positive and when it vanishes in a subdomain, we
completely discuss the issues of existence and uniqueness of
positive solutions, as well as their behavior with respect to the
parameter $\lambda$.

CPAA

We study a system of two porous medium type equations in a bounded
interval, coupled at the boundary in a nonlinear way. Under
certain conditions, one of its components becomes unbounded in
finite time while the other remains bounded, a situation that is
known in the literature as non-simultaneous blow-up. We
characterize completely, in the case of nondecreasing in time
solutions, the set of parameters appearing in the system for which
non-simultaneous blow-up indeed occurs. Moreover, we obtain the
blow-up rate and the blow-up set for the component which blows up.
We also prove that in the range of exponents where each of the
components may blow up on its own there are special initial data
such that blow-up is simultaneous. Finally, we give conditions on
the exponents which lead to non-simultaneous blow-up for every
initial data.

DCDS

In this paper we study existence and multiplicity of nonnegative
solutions to
$$
\begin{equation}
\left\{\begin{array}{ll}
\Delta u = u^p + u^q \qquad & \mbox{in }\Omega, \\
\frac{\partial u }{\partial \nu} =\lambda u \qquad & \mbox{on }\partial \Omega.
\end{array}\right.
\end{equation}
$$
Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\nu$ stands for
the outward unit normal and $p$, $q$ are in the convex-concave
case, that is $0 < q < 1 < p$. We prove that there exists $\Lambda^* >0$
such that there are no nonnegative solutions for $\lambda <
\Lambda^*$, and there is a maximal nonnegative solution for $\lambda
\ge \Lambda^{*}$. If $\lambda$ is large enough, then there exist at least
two nonnegative solutions. We also study the asymptotic behavior
of solutions when $\lambda\to \infty$ and the occurrence of dead
cores. In the particular case where $\Omega$ is the unit ball of
$\mathbb{R}^N$ we show exact multiplicity of radial nonnegative solutions
when $\lambda$ is large enough, and also the existence of nonradial
nonnegative solutions.

CPAA

We consider the obstacle problem for the infinity Laplace equation.
Given a Lipschitz boundary function and a Lipschitz obstacle we prove the
existence and uniqueness of a super infinity-harmonic function constrained to lie above
the obstacle which is infinity harmonic where it lies strictly above the obstacle.
Moreover, we show that this function is the limit of
value functions of a game we call obstacle tug-of-war.

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