It is well known that for many semilinear parabolic equations there is a global attractor which has a cell complex structure with finite dimensional cells. Additionally, many semilinear parabolic equations have equilibria with finite dimensional unstable manifolds. In this article, these results are unified to show that for a specific parabolic equation on an unbounded domain, the space of hete...

Linear quasi-parabolic integro-differential equations and their control appear in many scientific problems and engineering applications such as biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, and visco-elasticity, etc.. The existence and uniqueness of the solution of the linear quasi-parabolic integro-differential equations have been studied by Wheeler M....

in this paper, a nonlinear inverse problem of parabolic type, is considered. by reducing this inverse problem to a system of volterra integral equations the existence, uniqueness, and stability of the solution will be shown.

The bidomain equations represent the most complete description of cardiac electrical activity. However, the equations prove computationally burdensome as the resulting system of equations has two entries per spatial node. This paper examines the computational performance obtained by decoupling the bidomain equations into two separate systems of equations, an elliptic equation for the extracellu...

Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Mathias Masson Name of the doctoral dissertation Regularity for parabolic quasiminimizers in metric measure spaces Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series DOCTORAL DISSERTATIONS 89/2013 Field of research Mathematical analysis Manuscript submitte...

CONTENTS Introduction 151 § 1. Maximal attractors of semigroups generated by evolution equations 156 § 2. Examples of parabolic equations and systems having a maximal attractor 158 § 3. The Hausdorff dimension of invariant sets 164 § 4. Estimate of the change in volume under the action of shift operators generated by linear evolution equations 167 § 5. An upper bound for the Hausdorff dimension...

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, ...

We address the self-similar solvability of a singular parabolic problem and show that solutions to parabolic fully nonlinear equations are not expected to be C.