22 research outputs found
Global existence of weak solutions for strongly damped wave equations with nonlinear boundary conditions and balanced potentials
We demonstrate the global existence of weak solutions to a class of
semilinear strongly damped wave equations possessing nonlinear hyperbolic
dynamic boundary conditions. Our work assumes
with and where is the Wentzell-Laplacian.
Hence, the associated linear operator admits a compact resolvent. A balance
condition is assumed to hold between the nonlinearity defined on the interior
of the domain and the nonlinearity on the boundary. This allows for arbitrary
(supercritical) polynomial growth on each potential, as well as mixed
dissipative/anti-dissipative behavior. Moreover, the nonlinear function defined
on the interior of the domain is assumed to be only
Attractors for Damped Semilinear Wave Equations with Singularly Perturbed Acoustic Boundary Conditions
Under consideration is the damped semilinear wave equation in a bounded domain in
subject to an acoustic boundary condition with a singular perturbation, which
we term "massless acoustic perturbation," \ep\delta_{tt}+\delta_t+\delta =
-u_t\quad\text{for}\quad \ep\in[0,1]. By adapting earlier work by S.
Frigeri, we prove the existence of a family of global attractors for each
\ep\in[0,1]. We also establish the optimal regularity for the global
attractors, as well as the existence of an exponential attractor, for each
\ep\in[0,1]. The later result insures the global attractors possess finite
(fractal) dimension, however, we cannot yet guarantee that this dimension is
independent of the perturbation parameter \ep. The family of global
attractors are upper-semicontinuous with respect to the perturbation parameter
\ep, a result which follows by an application of a new abstract result also
contained in this article. Finally, we show that it is possible to obtain the
global attractors using weaker assumptions on the nonlinear term , however,
in that case, the optimal regularity, the finite dimensionality, and the
upper-semicontinuity of the global attractors does not necessarily hold.Comment: To appear in EJDE. arXiv admin note: substantial text overlap with
arXiv:1503.01821 and text overlap with arXiv:1302.426
Hyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions
Under consideration is the hyperbolic relaxation of a semilinear
reaction-diffusion equation on a bounded domain, subject to a dynamic boundary
condition. We also consider the limit parabolic problem with the same dynamic
boundary condition. Each problem is well-posed in a suitable phase space where
the global weak solutions generate a Lipschitz continuous semiflow which admits
a bounded absorbing set. We prove the existence of a family of global
attractors of optimal regularity. After fitting both problems into a common
framework, a proof of the upper-semicontinuity of the family of global
attractors is given as the relaxation parameter goes to zero. Finally, we also
establish the existence of exponential attractors.Comment: to appear in Quarterly of Applied Mathematic
Modeling change in public sentiment with nonlocal reaction-diffusion equations
This is a brief "proof of concept" article that shows nonlocal diffusion is
well suited to the study of pattern formation and the particular application of
public sentiment. We use a nonlocal reaction-diffusion equation to model the
evolution of public sentiment in a population that interacts with other
individuals. We employ a pseudo-random convolution kernel as a symmetric matrix
of lognormally distributed values. This kernel models the influence of
individuals when interacting with others. Change in sentiment emerges and may
converge to a polarized state. Other more complicated states occur whereby a
mixed polarization emerges.Comment: Keywords: Nonlocal diffusion; pattern formation; public sentimen