59 research outputs found
Exponential Energy Decay for Damped Klein-Gordon Equation with Nonlinearities of Arbitrary Growth
We derive a uniform exponential decay of the total energy for the nonlinear
Klein-Gordon equation with a damping around spatial infinity in the whole space
or in the exterior of a star shaped obstacle
A note on a class of 4th order hyperbolic problems with weak and strong damping and superlinear source term
In this paper we study a initial-boundary value problem for 4th order hyperbolic equations with weak and strong damping terms and superlinear source term. For blow-up solutions a lower bound of the blow-up time is derived. Then we extend the results to a class of equations where a positive power of gradient term is introduced
Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound
We consider an initial-boundary-value problem for a thermoelastic Kirchhoff &
Love plate, thermally insulated and simply supported on the boundary,
incorporating rotational inertia and a quasilinear hypoelastic response, while
the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law
giving rise to a 'second sound' effect. We study the local well-posedness of
the resulting quasilinear mixed-order hyperbolic system in a suitable solution
class of smooth functions mapping into Sobolev -spaces. Exploiting the
sole source of energy dissipation entering the system through the hyperbolic
heat flux moment, provided the initial data are small in a lower topology
(basic energy level corresponding to weak solutions), we prove a nonlinear
stabilizability estimate furnishing global existence & uniqueness and
exponential decay of classical solutions.Comment: 46 page
Boundary Stabilization of Quasilinear Maxwell Equations
We investigate an initial-boundary value problem for a quasilinear
nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary
condition of Silver & M\"uller type in a smooth, bounded, strictly star-shaped
domain of . Imposing usual smallness assumptions in addition to
standard regularity and compatibility conditions, a nonlinear stabilizability
inequality is obtained by showing nonlinear dissipativity and
observability-like estimates enhanced by an intricate regularity analysis. With
the stabilizability inequality at hand, the classic nonlinear barrier method is
employed to prove that small initial data admit unique classical solutions that
exist globally and decay to zero at an exponential rate. Our approach is based
on a recently established local well-posedness theory in a class of
-valued functions.Comment: 22 page
Exponential Decay of Quasilinear Maxwell Equations with Interior Conductivity
We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a
bounded smooth domain of with a strictly positive conductivity
subject to the boundary conditions of a perfect conductor. Under appropriate
regularity conditions, adopting a classical -Sobolev solution framework,
a nonlinear energy barrier estimate is established for local-in-time
-solutions to the Maxwell system by a proper combination of higher-order
energy and observability-type estimates under a smallness assumption on the
initial data. Technical complications due to quasilinearity, anisotropy and the
lack of solenoidality, etc., are addressed. Finally, provided the initial data
are small, the barrier method is applied to prove that local solutions exist
globally and exhibit an exponential decay rate.Comment: 24 page
On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions
In this paper we analyze the porous medium equation
\begin{equation}\label{ProblemAbstract} \tag{} %\begin{cases}
u_t=\Delta u^m + a\io u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad
\textrm{in}\quad \Omega \times I,%\\ %u_\nu-g(u)=0 & \textrm{on}\; \partial
\Omega, t>0,\\ %u({\bf x},0)=u_0({\bf x})&{\bf x} \in \Omega,\\ %\end{cases}
\end{equation} where is a bounded and smooth domain of , with
, and is the maximal interval of existence for . The
constants are positive, proper real numbers larger than 1 and
the equation is complemented with nonlinear boundary conditions involving the
outward normal derivative of . Under some hypothesis on the data, including
intrinsic relations between and , and assuming that for some positive
and sufficiently regular function u_0(\nx) the Initial Boundary Value Problem
(IBVP) associated to \eqref{ProblemAbstract} possesses a positive classical
solution u=u(\nx,t) on : \begin{itemize} \item
[] when and in 2- and 3-dimensional domains, we determine
a \textit{lower bound of} for those becoming unbounded in
at such ; \item [] when and in
-dimensional settings, we establish a \textit{global existence criterion}
for . \end{itemize
Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping
AbstractWe consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
São Carlos, Brasil. 3-7 february 2014
Long-time behavior of quasilinear thermoelastic Kirchhoff-Love plates with second sound
We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a `second sound\u27 effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev -spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small in a lower topology (basic energy level corresponding to weak solutions), we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions
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