15 citations · 36 across the 15 of their papers we have counts for
16 papers
Backward Reconstruction of the Chafee--Infante Equation via Physics-Informed WGAN-GP
Joseph L. Shomberg
We present a physics-informed Wasserstein GAN with gradient penalty (WGAN-GP) for solving the inverse Chafee--Infante problem on two-dimensional domains with Dirichlet boundary con…
Modeling change in public sentiment with nonlocal reaction-diffusion equations
Joseph L. Shomberg
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…
Weak Exponential Attractors for Coleman--Gurtin Equations with Dynamic Boundary Conditions Possessing Different Memory Kernels
Joseph L. Shomberg
The well-posedness of a generalized Coleman--Gurtin equation equipped with dynamic boundary conditions with memory was recently established by C.G. Gal and the author. Additionally…
Global existence of weak solutions for strongly damped wave equations with nonlinear boundary conditions and balanced potentials
Joseph L. Shomberg
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…
Upper-semicontinuity of the global attractors for a class of nonlocal Cahn-Hilliard equations
Joseph L. Shomberg
The aim of this work is to examine the upper-semicontinuity properties of the family of global attractors admitted by a non-isothermal viscous relaxation of some nonlocal Cahn-Hill…
Well-posedness and Global Attractors for Viscous Fractional Cahn-Hilliard Equations with Memory
Eylem Öztürk, Joseph L. Shomberg
We examine a viscous Cahn-Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global w…
Well-posedness and global attractors for a non-isothermal viscous relaxation of nonlocal Cahn-Hilliard equations
Joseph L. Shomberg
We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators, exhibiting dissipat…
Regular global attractors for wave equations with degenerate memory
Joseph L. Shomberg
We consider the wave equation with degenerate viscoelastic dissipation recently examined in Cavalcanti, Fatori, and Ma, Attractors for wave equations with degenerate memory, J. Dif…
Robust Exponential Attractors for Coleman--Gurtin Equations with Dynamic Boundary Conditions Possessing Memory
Joseph L. Shomberg
The well-posedness of a generalized Coleman--Gurtin equation equipped with dynamic boundary conditions with memory was recently established by the author with C.G. Gal. In this art…
Attractors for Damped Semilinear Wave Equations with Singularly Perturbed Acoustic Boundary Conditions
Joseph L. Shomberg
Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-Δu+u+f(u)=0 \] in a bounded domain in subject to an acoustic boundary condition with a s…
Attractors for Strongly Damped Wave Equations with Nonlinear Hyperbolic Dynamic Boundary Conditions
P. Jameson Graber, Joseph L. Shomberg
We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the prese…
Attractors for Damped Semilinear Wave Equations with a Robin--Acoustic Boundary Perturbation
Joseph L. Shomberg
Under consideration is the damped semilinear wave equation \[ u_{tt}+u_t-Δu + u + f(u)=0 \] on a bounded domain in with a perturbation parameter …
Coleman-Gurtin type equations with dynamic boundary conditions
Ciprian G. Gal, Joseph L. Shomberg
We present a new formulation and generalization of the classical theory of heat conduction with or without fading memory which includes the usual heat equation subject to a dynamic…
Exponential Decay Results for Semilinear Parabolic PDE with Potentials: A "Mean Value" Approach
Joseph L. Shomberg
The asymptotic behavior of some semilinear parabolic PDEs is analyzed by means of a "mean value" property. This property allows us to determine, by means of appropriate {\em{a prio…
Hyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions
Ciprian G. Gal, Joseph L. Shomberg
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 limi…
Explicit Construction of a Robust Family of Compact Inertial Manifolds
Joseph Lee Shomberg
A construction of a robust family of compact inertial manifolds is presented. The result aims to complete an analysis of certain types of attracting sets for a class of dissipative…